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PROFESSOR: OK.

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00:00:25,230 --> 00:00:28,710
Today we're all through with
Markov chains, or at least

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00:00:28,710 --> 00:00:32,530
with finite state
Markov chains.

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00:00:32,530 --> 00:00:36,440
And we're going on to
renewal processes.

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00:00:36,440 --> 00:00:40,660
As part of that, we will spend
a good deal of time talking

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00:00:40,660 --> 00:00:44,950
about the strong law of large
numbers, and convergence with

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00:00:44,950 --> 00:00:46,830
probability one.

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00:00:46,830 --> 00:00:51,440
The idea of convergence with
probability one, at least to

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00:00:51,440 --> 00:00:56,540
me is by far the most difficult
part of the course.

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00:00:56,540 --> 00:00:59,890
It's very abstract
mathematically.

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00:00:59,890 --> 00:01:03,170
It looks like it's simple, and
it's one of those things you

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00:01:03,170 --> 00:01:06,630
start to think you understand
it, and then at a certain

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00:01:06,630 --> 00:01:09,460
point, you realize
that you don't.

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00:01:09,460 --> 00:01:12,740
And this has been happening
to me for 20 years now.

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00:01:12,740 --> 00:01:16,650
I keep thinking I really
understand this idea of

23
00:01:16,650 --> 00:01:19,800
convergence with probability
one, and then I see some

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00:01:19,800 --> 00:01:22,180
strange example again.

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00:01:22,180 --> 00:01:25,130
And I say there's something
very peculiar

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00:01:25,130 --> 00:01:26,500
about this whole idea.

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00:01:26,500 --> 00:01:29,750
And I'm going to illustrate
that you at the end of the

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00:01:29,750 --> 00:01:30,980
lecture today.

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00:01:30,980 --> 00:01:35,100
But for the most part, I will
be talking not so much about

30
00:01:35,100 --> 00:01:40,260
renewal processes, but this
set of mathematical issues

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00:01:40,260 --> 00:01:43,990
that we have to understand in
order to be able to look at

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00:01:43,990 --> 00:01:47,440
renewal processes in
the simplest way.

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00:01:47,440 --> 00:01:50,830
One of the funny things about
the strong law of large

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00:01:50,830 --> 00:01:55,420
numbers and how it gets applied
to renewal processes

35
00:01:55,420 --> 00:02:01,220
is that although the idea of
convergence with probability

36
00:02:01,220 --> 00:02:06,990
one is sticky and strange, once
you understand it, it is

37
00:02:06,990 --> 00:02:11,920
one of the most easy things
to use there is.

38
00:02:11,920 --> 00:02:16,220
And therefore, once you become
comfortable with it, you can

39
00:02:16,220 --> 00:02:19,150
use it to do things which would
be very hard to do in

40
00:02:19,150 --> 00:02:21,260
any other way.

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00:02:21,260 --> 00:02:24,630
And because of that, most people
feel they understand it

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00:02:24,630 --> 00:02:26,700
better than they actually do.

43
00:02:26,700 --> 00:02:30,670
And that's the reason why it
sometimes crops up when you're

44
00:02:30,670 --> 00:02:33,230
least expecting it, and
you find there's

45
00:02:33,230 --> 00:02:35,630
something very peculiar.

46
00:02:35,630 --> 00:02:39,050
OK, so let's start out by
talking a little bit about

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00:02:39,050 --> 00:02:41,580
renewal processes.

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00:02:41,580 --> 00:02:47,570
And then talking about this
convergence, and the strong

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00:02:47,570 --> 00:02:52,190
law of large numbers, and what
it does to all of this.

50
00:02:52,190 --> 00:02:54,090
This is just review.

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00:02:54,090 --> 00:02:57,230
We talked about arrival
processes when we started

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00:02:57,230 --> 00:03:00,090
talking about Poisson
processes.

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00:03:00,090 --> 00:03:03,740
Renewal processes are a special
kind of arrival

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00:03:03,740 --> 00:03:07,510
processes, and Poisson processes
are a special kind

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00:03:07,510 --> 00:03:09,290
of renewal process.

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00:03:09,290 --> 00:03:13,710
So this is something you're
already sort of familiar with.

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00:03:13,710 --> 00:03:20,310
All of arrival processes, we
will tend to treat in one of

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00:03:20,310 --> 00:03:23,450
three equivalent ways, which is
the same thing we did with

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00:03:23,450 --> 00:03:24,700
Poisson processes.

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00:03:27,250 --> 00:03:30,250
A stochastic process,
we said, is a

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00:03:30,250 --> 00:03:32,410
family of random variables.

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00:03:32,410 --> 00:03:38,070
But in this case, we always view
it as three families of

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00:03:38,070 --> 00:03:40,770
random variables, which
are all related.

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00:03:40,770 --> 00:03:43,720
And all of which define
the other.

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00:03:43,720 --> 00:03:46,780
And you jump back and forth from
looking at one to looking

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00:03:46,780 --> 00:03:51,650
at the other, which is, as you
saw with Poisson processes,

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00:03:51,650 --> 00:03:55,250
you really want to do this,
because if you stick to only

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00:03:55,250 --> 00:04:00,060
one way of looking at it, you
really only pick up about a

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00:04:00,060 --> 00:04:02,736
quarter, or a half
of the picture.

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00:04:02,736 --> 00:04:04,840
OK.

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00:04:04,840 --> 00:04:08,810
So this one picture gives us
a relationship between the

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00:04:08,810 --> 00:04:14,170
arrival epochs of an arrival
process, the inter-arrival

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00:04:14,170 --> 00:04:21,560
intervals, the x1, x2, x3, and
the counting process, n of t,

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00:04:21,560 --> 00:04:26,100
and whichever one you use, you
use the one which is easiest,

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00:04:26,100 --> 00:04:28,330
for whatever you plan to do.

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00:04:28,330 --> 00:04:31,980
For defining a renewal process,
the easy thing to do

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00:04:31,980 --> 00:04:36,330
is to look at the inter-arrival
intervals,

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00:04:36,330 --> 00:04:40,280
because the definition of a
renewal process is it's an

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00:04:40,280 --> 00:04:45,930
arrival process for which the
interrenewals are independent

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00:04:45,930 --> 00:04:48,290
and identically distributed.

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So any process where the
arrivals, inter-arrivals have

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00:04:53,000 --> 00:04:56,260
that property are IID.

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00:04:56,260 --> 00:05:00,640
OK, renewal processes are
characterized, and the name

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00:05:00,640 --> 00:05:06,700
comes from the idea that you
start over at each interval.

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00:05:06,700 --> 00:05:11,180
This idea of starting over is
something that we talk about

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00:05:11,180 --> 00:05:13,350
more later on.

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00:05:13,350 --> 00:05:17,310
And it's a little bit strange,
and a little bit fishy.

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00:05:17,310 --> 00:05:21,910
It's like with a Poisson
process, you look at different

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00:05:21,910 --> 00:05:25,400
intervals, and they're
independent of each other.

90
00:05:25,400 --> 00:05:28,860
And we sort of know what
that means by now.

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00:05:28,860 --> 00:05:34,540
OK, you look at the arrival
epochs for a Poisson process.

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00:05:34,540 --> 00:05:37,280
Are they independent
of each other?

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00:05:37,280 --> 00:05:39,210
Of course not.

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00:05:39,210 --> 00:05:41,450
The arrival epochs are
the sums of the

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00:05:41,450 --> 00:05:43,520
inter-arrival intervals.

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00:05:43,520 --> 00:05:45,950
The inter-arrival intervals
are the things that are

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00:05:45,950 --> 00:05:47,190
independent.

98
00:05:47,190 --> 00:05:49,260
And the arrival epochs
are the sums of

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00:05:49,260 --> 00:05:50,810
inter-arrival intervals.

100
00:05:50,810 --> 00:05:54,250
If you know that the first
arrival epoch takes 10 times

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00:05:54,250 --> 00:05:59,480
longer than its mean, then that
the second arrival epoch

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00:05:59,480 --> 00:06:01,590
is going to be kind
of long, too.

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00:06:01,590 --> 00:06:05,910
It's got to be at least 10 times
as long as the mean of

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00:06:05,910 --> 00:06:10,330
the inter-arrival epochs,
because each arrival epoch is

105
00:06:10,330 --> 00:06:13,130
a sum of these inter-arrival
epochs.

106
00:06:13,130 --> 00:06:15,980
It's the inter-arrival epochs
that are independent.

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00:06:15,980 --> 00:06:19,720
So when you say that the one
interval is independent of the

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00:06:19,720 --> 00:06:23,720
other, yes, you know exactly
what you mean.

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00:06:23,720 --> 00:06:26,160
And the idea is very simple.

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00:06:26,160 --> 00:06:28,890
It's the same idea here.

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00:06:28,890 --> 00:06:32,490
But then you start to think you
understand this, and you

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00:06:32,490 --> 00:06:35,980
start to use it in
a funny way.

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00:06:35,980 --> 00:06:39,710
And suddenly you're starting to
say that the arrival epochs

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00:06:39,710 --> 00:06:42,280
are independent from one time
to the other, which they

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00:06:42,280 --> 00:06:44,050
certainly aren't.

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00:06:44,050 --> 00:06:49,070
What renewal theory does is it
lets you treat the gross

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00:06:49,070 --> 00:06:53,000
characteristics of a process
in a very simple and

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00:06:53,000 --> 00:06:54,640
straightforward way.

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00:06:54,640 --> 00:06:58,270
So you're breaking up the
process into two sets

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00:06:58,270 --> 00:06:59,700
of views about it.

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00:06:59,700 --> 00:07:03,470
One is the long term behavior,
which you treat by renewal

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00:07:03,470 --> 00:07:10,110
theory, and you use this one
exotic theory in a simple and

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00:07:10,110 --> 00:07:14,550
straightforward way for every
different process, for every

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00:07:14,550 --> 00:07:17,150
different renewal process
you look at.

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00:07:17,150 --> 00:07:21,230
And then you have this usually
incredibly complicated kind of

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00:07:21,230 --> 00:07:26,260
thing in the inside of
each arrival epoch.

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00:07:26,260 --> 00:07:29,650
And the nice thing about renewal
theory is it lets you

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00:07:29,650 --> 00:07:33,180
look at that complicated thing
without worrying about what's

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00:07:33,180 --> 00:07:35,090
going on outside.

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00:07:35,090 --> 00:07:38,950
So the local characteristics
can be studied without

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00:07:38,950 --> 00:07:43,410
worrying about the long
term interactions.

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00:07:43,410 --> 00:07:48,470
One example of this, and one
of the reasons we are now

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00:07:48,470 --> 00:07:51,280
looking at Markov chains before
we look at renewal

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00:07:51,280 --> 00:07:57,000
processes is that a Markov chain
is one of the nicest

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00:07:57,000 --> 00:08:01,890
examples there is of a renewal
process, when you look at it

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00:08:01,890 --> 00:08:03,450
in the right way.

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00:08:03,450 --> 00:08:11,150
If you have a recurrent Markov
chain, then the interval from

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00:08:11,150 --> 00:08:15,360
one time entering a particularly
recurrent state

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00:08:15,360 --> 00:08:17,190
until the next time
you enter that

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00:08:17,190 --> 00:08:21,500
recurrent state is a renewal.

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00:08:21,500 --> 00:08:27,570
So we look at the sequence of
times at which we enter this

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00:08:27,570 --> 00:08:29,110
one given state.

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00:08:29,110 --> 00:08:31,150
Enter state one over here.

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00:08:31,150 --> 00:08:33,520
We enter state one
again over here.

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00:08:33,520 --> 00:08:36,780
We enter state one again,
and so forth.

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00:08:36,780 --> 00:08:39,710
We're ignoring everything
that goes on between

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00:08:39,710 --> 00:08:41,539
entries to state one.

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00:08:41,539 --> 00:08:44,810
But every time you enter state
1, you're in the same

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00:08:44,810 --> 00:08:49,490
situation as you were the last
time you entered state one.

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00:08:49,490 --> 00:08:53,150
You're in the same situation,
in the sense that the

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00:08:53,150 --> 00:08:58,010
inter-arrivals from state one
to state one again are

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00:08:58,010 --> 00:08:59,910
independent of what
they were before.

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00:08:59,910 --> 00:09:03,780
In other words, when you enter
state one, your successive

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00:09:03,780 --> 00:09:07,650
state transitions from
there are the same

155
00:09:07,650 --> 00:09:08,850
as they were before.

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00:09:08,850 --> 00:09:15,500
So it's the same situation as we
saw with Poisson processes,

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00:09:15,500 --> 00:09:19,930
and it's the same kind of
renewal where when you talk

158
00:09:19,930 --> 00:09:25,800
about renewal, you have to be
very careful about what it is

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00:09:25,800 --> 00:09:26,740
that's a renewable.

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00:09:26,740 --> 00:09:30,420
Once you're careful about it,
it's clear what's going on.

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00:09:30,420 --> 00:09:33,000
One of the things we're going to
find out now is one of the

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00:09:33,000 --> 00:09:36,610
things that we failed to point
out before when we talked

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00:09:36,610 --> 00:09:39,400
about finite state and
Markov chains.

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00:09:39,400 --> 00:09:43,080
One of the most interesting
characteristics is the

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00:09:43,080 --> 00:09:47,620
expected amount of time from one
entry to a recurrent state

166
00:09:47,620 --> 00:09:51,380
until the next time you enter
that recurrent state is 1 over

167
00:09:51,380 --> 00:09:55,390
pi sub i, where pi sub i is a
steady state probability of

168
00:09:55,390 --> 00:09:57,670
that steady state.

169
00:09:57,670 --> 00:09:59,600
Namely, we didn't do that.

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00:09:59,600 --> 00:10:03,290
It's a little tricky to do that
in terms of Markov chains

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00:10:03,290 --> 00:10:07,530
it's almost trivial to do it in
terms of renewal processes.

172
00:10:07,530 --> 00:10:11,450
And what's more, when we do it
in terms of renewal processes,

173
00:10:11,450 --> 00:10:13,230
you will see that it's
obvious, and you will

174
00:10:13,230 --> 00:10:14,410
never forget it.

175
00:10:14,410 --> 00:10:17,120
If we did it in terms of Markov
chains, it would be

176
00:10:17,120 --> 00:10:21,700
some long, tedious derivation,
and you'd get this nice

177
00:10:21,700 --> 00:10:24,970
answer, and you say, why did
that nice answer occur?

178
00:10:24,970 --> 00:10:26,500
And you wouldn't
have any idea.

179
00:10:26,500 --> 00:10:29,000
When you look at renewal
processes, it's

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00:10:29,000 --> 00:10:31,100
obvious why it happens.

181
00:10:31,100 --> 00:10:34,130
And we'll see why that
is very soon.

182
00:10:34,130 --> 00:10:38,310
Also, after we finish renewal
processes, the next thing

183
00:10:38,310 --> 00:10:41,330
we're going to do is to
talk about accountable

184
00:10:41,330 --> 00:10:43,090
state Markov chains.

185
00:10:43,090 --> 00:10:46,740
Markov chains with accountable,
infinitely

186
00:10:46,740 --> 00:10:48,025
countable set of states.

187
00:10:52,520 --> 00:10:56,100
If you don't have a background
in renewal theory when you

188
00:10:56,100 --> 00:10:59,600
start to look at that, you
get very confused.

189
00:10:59,600 --> 00:11:02,940
So renewal theory will give us
the right tool to look at

190
00:11:02,940 --> 00:11:07,140
those more complicated
Markov chains.

191
00:11:07,140 --> 00:11:07,830
OK.

192
00:11:07,830 --> 00:11:10,620
So we carry from Markov chains
with accountably infinite

193
00:11:10,620 --> 00:11:14,940
state space comes largely
from renewal process.

194
00:11:14,940 --> 00:11:19,350
So yes, we'll be interested
in understanding that.

195
00:11:19,350 --> 00:11:19,750
OK.

196
00:11:19,750 --> 00:11:25,430
Another example is GTM queue.

197
00:11:25,430 --> 00:11:28,980
The text talked a little bit,
and we might have talked in

198
00:11:28,980 --> 00:11:32,810
class a little bit about this
strange notation a queueing

199
00:11:32,810 --> 00:11:34,710
theorist used.

200
00:11:34,710 --> 00:11:37,360
There are always at least three
letters separated by

201
00:11:37,360 --> 00:11:41,140
slashes to talk about
what kind of queue

202
00:11:41,140 --> 00:11:42,670
you're talking about.

203
00:11:42,670 --> 00:11:47,990
The first letter describes the
arrival process for the queue.

204
00:11:47,990 --> 00:11:52,860
G means it's a general rival
process, which doesn't really

205
00:11:52,860 --> 00:11:54,960
mean it's a general
arrival process.

206
00:11:54,960 --> 00:11:58,210
It means the arrival
process is renewal.

207
00:11:58,210 --> 00:12:03,210
Namely, it says the arrival
process is IID,

208
00:12:03,210 --> 00:12:05,280
inter-arrivals.

209
00:12:05,280 --> 00:12:07,890
But you don't know what
their distribution is.

210
00:12:07,890 --> 00:12:11,060
You would call that M if you
meant a Poisson process, which

211
00:12:11,060 --> 00:12:14,780
would mean memory lists,
inter-arrivals.

212
00:12:14,780 --> 00:12:19,940
The second G stands for the
service time distribution.

213
00:12:19,940 --> 00:12:23,690
Again, we assume that no matter
how many servers you

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00:12:23,690 --> 00:12:27,700
have, no matter how the servers
work, the time to

215
00:12:27,700 --> 00:12:31,980
serve one user is independent
of the time

216
00:12:31,980 --> 00:12:33,600
to serve other users.

217
00:12:33,600 --> 00:12:37,370
But that the distribution of
that time has a general

218
00:12:37,370 --> 00:12:38,600
distribution.

219
00:12:38,600 --> 00:12:42,800
It would be M as you meant a
memory list distribution,

220
00:12:42,800 --> 00:12:45,680
which would mean exponential
distribution.

221
00:12:45,680 --> 00:12:51,110
Finally, the thing at the end
says we're talking about IQ

222
00:12:51,110 --> 00:12:52,950
with M servers.

223
00:12:52,950 --> 00:12:55,180
So the point here is we're
talking about a relatively

224
00:12:55,180 --> 00:12:57,690
complicated thing.

225
00:12:57,690 --> 00:13:00,550
Can you talk about this
in terms of renewals?

226
00:13:00,550 --> 00:13:05,290
Yes, you can, but it's not quite
obvious how to do it.

227
00:13:05,290 --> 00:13:09,550
You would think that the obvious
way of viewing a

228
00:13:09,550 --> 00:13:14,620
complicated queue like this is
to look at what happens from

229
00:13:14,620 --> 00:13:18,040
one busy period to the
next busy period.

230
00:13:18,040 --> 00:13:20,590
You would think the busy periods
would be independent

231
00:13:20,590 --> 00:13:21,800
of each other.

232
00:13:21,800 --> 00:13:23,610
But they're not quite.

233
00:13:23,610 --> 00:13:27,820
Suppose you finish one busy
period, and when you finish

234
00:13:27,820 --> 00:13:33,525
the one busy period, one
customer has just finished

235
00:13:33,525 --> 00:13:34,930
being served.

236
00:13:34,930 --> 00:13:38,630
But at that point, you're in the
middle of the waiting for

237
00:13:38,630 --> 00:13:41,230
the next customer to arrive.

238
00:13:41,230 --> 00:13:43,630
And as that's a general
distribution, the amount of

239
00:13:43,630 --> 00:13:47,380
time you have to wait for that
next customer to arrive

240
00:13:47,380 --> 00:13:50,120
depends on a whole
lot of things in

241
00:13:50,120 --> 00:13:52,000
the previous interval.

242
00:13:52,000 --> 00:13:54,540
So how can you talk about
renewals here?

243
00:13:54,540 --> 00:13:57,520
You talk about renewables by
waiting until that next

244
00:13:57,520 --> 00:13:58,730
arrival comes.

245
00:13:58,730 --> 00:14:04,270
When that next arrival comes to
terminate the idle period

246
00:14:04,270 --> 00:14:07,970
between busy periods, at that
time you're in the same state

247
00:14:07,970 --> 00:14:10,730
that you were in when the whole
thing started before.

248
00:14:10,730 --> 00:14:14,180
When you had the first
arrival come in.

249
00:14:14,180 --> 00:14:17,480
And at that point, you had one a
rival there being served you

250
00:14:17,480 --> 00:14:20,360
go through some long
complicated thing.

251
00:14:20,360 --> 00:14:22,730
Eventually the busy
period is over.

252
00:14:22,730 --> 00:14:25,760
Eventually, then, another
arrival comes in.

253
00:14:25,760 --> 00:14:28,920
And presto, at that point,
you're statistically back

254
00:14:28,920 --> 00:14:30,070
where you started.

255
00:14:30,070 --> 00:14:33,560
You're statistically back where
you started in terms of

256
00:14:33,560 --> 00:14:37,400
all inter-arrival times
at that point.

257
00:14:37,400 --> 00:14:40,220
And we will have to, even
though it's intuitively

258
00:14:40,220 --> 00:14:44,300
obvious that those things are
independent of each other,

259
00:14:44,300 --> 00:14:47,380
we're really going to have to
sort that out a little bit,

260
00:14:47,380 --> 00:14:51,290
because you come upon
many situations

261
00:14:51,290 --> 00:14:53,820
where this is not obvious.

262
00:14:53,820 --> 00:14:56,310
So if you don't know how to
sort it out when it is

263
00:14:56,310 --> 00:14:59,480
obvious, you're not going to
know how to sort it out when

264
00:14:59,480 --> 00:15:00,490
it's not obvious.

265
00:15:00,490 --> 00:15:03,840
But anyway, that's another
example of

266
00:15:03,840 --> 00:15:06,650
where we have renewals.

267
00:15:06,650 --> 00:15:08,710
OK.

268
00:15:08,710 --> 00:15:11,760
We want to talk about
convergence now.

269
00:15:11,760 --> 00:15:15,170
This idea of convergence
with probability one.

270
00:15:15,170 --> 00:15:17,990
It's based on the
idea of numbers

271
00:15:17,990 --> 00:15:21,440
converging to some limit.

272
00:15:21,440 --> 00:15:27,430
And I'm always puzzled about how
much to talk about this,

273
00:15:27,430 --> 00:15:31,440
because all of you, when you
first study calculus, talk

274
00:15:31,440 --> 00:15:32,990
about limits.

275
00:15:32,990 --> 00:15:36,650
Most of you, if you're
engineers, when you talk about

276
00:15:36,650 --> 00:15:40,400
calculus, it goes in one ear and
it goes out the other ear,

277
00:15:40,400 --> 00:15:43,900
because you don't have to
understand this very much.

278
00:15:43,900 --> 00:15:46,820
Because all the things you deal
with, the limits exist

279
00:15:46,820 --> 00:15:49,330
very nicely, and there's
no problem.

280
00:15:49,330 --> 00:15:51,170
So you can ignore it.

281
00:15:51,170 --> 00:15:55,310
And then you hear about these
epsilons and deltas, and I do

282
00:15:55,310 --> 00:15:56,930
the same thing.

283
00:15:56,930 --> 00:15:59,840
I can deal with an epsilon,
but as soon as you have an

284
00:15:59,840 --> 00:16:03,890
epsilon and a delta,
I go into orbit.

285
00:16:03,890 --> 00:16:07,210
I have no idea what's going on
anymore until I sit down and

286
00:16:07,210 --> 00:16:09,350
think about it very,
very carefully.

287
00:16:09,350 --> 00:16:13,200
Fortunately, when we have a
sequence of numbers, we only

288
00:16:13,200 --> 00:16:14,060
have an epsilon.

289
00:16:14,060 --> 00:16:15,610
We don't have a delta.

290
00:16:15,610 --> 00:16:17,095
So things are a little
bit simpler.

291
00:16:19,860 --> 00:16:23,260
I should warn you, though, that
you can't let this go in

292
00:16:23,260 --> 00:16:27,250
one ear and out the other ear,
because at this point, we are

293
00:16:27,250 --> 00:16:30,590
using the convergence of numbers
to be able to talk

294
00:16:30,590 --> 00:16:34,330
about convergence of random
variables, and convergence of

295
00:16:34,330 --> 00:16:38,500
random variables is indeed
not a simple topic.

296
00:16:38,500 --> 00:16:43,540
Convergence of numbers is a
simple topic made complicated

297
00:16:43,540 --> 00:16:44,790
by mathematicians.

298
00:16:46,800 --> 00:16:49,760
Any good mathematician,
when they hear me say

299
00:16:49,760 --> 00:16:51,430
this will be furious.

300
00:16:51,430 --> 00:16:55,210
Because in fact, when you think
about what they've done,

301
00:16:55,210 --> 00:16:59,080
they've taken something which
is simple but looks

302
00:16:59,080 --> 00:17:03,820
complicated, and they've turned
it into something which

303
00:17:03,820 --> 00:17:06,609
looks complicated in another
way, but is really the

304
00:17:06,609 --> 00:17:08,359
simplest way to deal with it.

305
00:17:08,359 --> 00:17:12,680
So let's do that and be done
with it, and then we can start

306
00:17:12,680 --> 00:17:15,140
using it for random variables.

307
00:17:15,140 --> 00:17:19,880
A sequence, b1, b2, b3, so
forth of real numbers.

308
00:17:19,880 --> 00:17:22,520
Real numbers are complex numbers
that doesn't make any

309
00:17:22,520 --> 00:17:30,640
difference, is said to converge
to a limit, b.

310
00:17:30,640 --> 00:17:33,470
If for each real epsilon greater
than zero, is there an

311
00:17:33,470 --> 00:17:37,700
integer M such that bn minus
b is less than or equal to

312
00:17:37,700 --> 00:17:41,460
epsilon for all n greater
than or equal to n?

313
00:17:41,460 --> 00:17:44,580
Now, how many people can look
at that and understand it?

314
00:17:44,580 --> 00:17:45,510
Be honest.

315
00:17:45,510 --> 00:17:46,250
Good.

316
00:17:46,250 --> 00:17:48,410
Some of you can.

317
00:17:48,410 --> 00:17:53,760
How many people look at that,
and their mind just, ah!

318
00:17:53,760 --> 00:17:57,010
How many people are
in that category?

319
00:17:57,010 --> 00:17:58,020
I am.

320
00:17:58,020 --> 00:18:00,660
But if I'm the only
one, that's good.

321
00:18:00,660 --> 00:18:03,070
OK.

322
00:18:03,070 --> 00:18:06,250
There's an equivalent way
to talk about this.

323
00:18:06,250 --> 00:18:10,110
A sequence of numbers, real or
complex, is said to converge

324
00:18:10,110 --> 00:18:11,420
to limit b.

325
00:18:11,420 --> 00:18:16,480
If for each integer k greater
than zero, there's an integer

326
00:18:16,480 --> 00:18:21,450
m of k, such that bn minus b
is less than or equal to 1

327
00:18:21,450 --> 00:18:25,825
over k for all n greater
than or equal to m.

328
00:18:25,825 --> 00:18:26,210
OK.

329
00:18:26,210 --> 00:18:30,160
And the argument there is pick
any epsilon you want to, no

330
00:18:30,160 --> 00:18:32,670
matter how small.

331
00:18:32,670 --> 00:18:35,930
And then you pick a k, such that
1 over k is less than or

332
00:18:35,930 --> 00:18:37,340
equal to epsilon.

333
00:18:37,340 --> 00:18:42,890
According to this definition, bn
minus b less than or equal

334
00:18:42,890 --> 00:18:48,990
to 1 over k ensures that you
have this condition up here

335
00:18:48,990 --> 00:18:51,120
that we're talking about.

336
00:18:51,120 --> 00:18:56,160
When bn minus b is less than
or equal to 1 over k, then

337
00:18:56,160 --> 00:18:59,950
also bn minus b is less than
or equal to epsilon.

338
00:18:59,950 --> 00:19:02,780
In other words, when you look
at this, you're starting to

339
00:19:02,780 --> 00:19:06,640
see what this definition
really means.

340
00:19:06,640 --> 00:19:12,700
Here, you don't really care
about all epsilon.

341
00:19:12,700 --> 00:19:18,200
All you care about is that
this holds true for small

342
00:19:18,200 --> 00:19:19,740
enough epsilon.

343
00:19:19,740 --> 00:19:23,890
And the trouble is there's
no way to specify a

344
00:19:23,890 --> 00:19:25,570
small enough epsilon.

345
00:19:25,570 --> 00:19:29,670
So the only way we can do this
is to say for all epsilon.

346
00:19:29,670 --> 00:19:35,700
But what the argument is is if
you can assert this statement

347
00:19:35,700 --> 00:19:39,650
for a sequence of smaller and
smaller values of epsilon,

348
00:19:39,650 --> 00:19:41,110
that's all you need.

349
00:19:41,110 --> 00:19:45,510
Because as soon as this is true
for one value of epsilon,

350
00:19:45,510 --> 00:19:48,200
it's true for all smaller
values of epsilon.

351
00:19:48,200 --> 00:19:53,110
Now, let me show you a picture
which, unfortunately, there's

352
00:19:53,110 --> 00:19:56,680
a kind of a complicated
picture.

353
00:19:56,680 --> 00:20:00,900
It's the picture that says what
that argument was really

354
00:20:00,900 --> 00:20:02,410
talking about.

355
00:20:02,410 --> 00:20:05,720
So if you don't understand the
picture, you were kidding

356
00:20:05,720 --> 00:20:08,230
yourself when you said you
thought you understood what

357
00:20:08,230 --> 00:20:10,090
the definition said.

358
00:20:10,090 --> 00:20:12,970
So what the picture says,
it's in terms of

359
00:20:12,970 --> 00:20:16,800
this 1 over k business.

360
00:20:16,800 --> 00:20:21,610
It says if you have a sequence
of numbers, b1, b2, b3, excuse

361
00:20:21,610 --> 00:20:24,220
me for insulting you
by talking about

362
00:20:24,220 --> 00:20:26,302
something so trivial.

363
00:20:26,302 --> 00:20:29,350
But believe me, as soon as we
start talking about random

364
00:20:29,350 --> 00:20:33,500
variables, this trivial thing
mixed with so many other

365
00:20:33,500 --> 00:20:37,160
things will start to become less
trivial, and you really

366
00:20:37,160 --> 00:20:40,320
need to understand what
this is saying.

367
00:20:40,320 --> 00:20:46,020
So we're saying if we have a
sequence, b1, b2, b3, b4, b5

368
00:20:46,020 --> 00:20:52,030
and so forth, what that second
idea of convergence says is

369
00:20:52,030 --> 00:21:02,760
that there's an M1 which says
that for all n greater than or

370
00:21:02,760 --> 00:21:12,500
equal to M1, b4, b5, b6, b7
minus b all lies within this

371
00:21:12,500 --> 00:21:16,660
limit here between b plus
1 and b minus 1.

372
00:21:16,660 --> 00:21:20,360
There's a number M2, which says
that as soon as you get

373
00:21:20,360 --> 00:21:23,570
bigger than M of 2, all
these numbers lie

374
00:21:23,570 --> 00:21:25,390
between these two limits.

375
00:21:25,390 --> 00:21:28,890
There's a number M3, which says
all of these numbers lie

376
00:21:28,890 --> 00:21:30,130
between these limits.

377
00:21:30,130 --> 00:21:34,670
So it's saying that, it's
essentially saying that you

378
00:21:34,670 --> 00:21:40,340
can a pipe, and as n increases,
you squeeze this

379
00:21:40,340 --> 00:21:42,000
pipe gradually down.

380
00:21:42,000 --> 00:21:44,550
You don't know how fast you
can squeeze it down when

381
00:21:44,550 --> 00:21:46,440
you're talking about
convergence.

382
00:21:46,440 --> 00:21:49,460
You might have something that
converges very slowly, and

383
00:21:49,460 --> 00:21:51,790
then M1 will be way out here.

384
00:21:51,790 --> 00:21:53,700
M2 will be way over there.

385
00:21:53,700 --> 00:21:58,480
M3 will be off on the
other side of Vassar

386
00:21:58,480 --> 00:22:01,160
Street, and so forth.

387
00:22:01,160 --> 00:22:05,650
But there always is such an
M1, M2, and M3, which says

388
00:22:05,650 --> 00:22:09,070
these numbers are getting
closer and closer to b.

389
00:22:09,070 --> 00:22:12,280
And they're staying closer
and closer to b.

390
00:22:12,280 --> 00:22:16,800
An example, which we'll come
back to, where you don't have

391
00:22:16,800 --> 00:22:20,840
convergence is the following
kind of thing.

392
00:22:20,840 --> 00:22:25,030
b1 is equal to 3/4,
in this case.

393
00:22:25,030 --> 00:22:27,470
b5 is equal to 3/4.

394
00:22:27,470 --> 00:22:30,110
b25 is equal to 3/4.

395
00:22:30,110 --> 00:22:34,540
b5 to the third is equal to 1.

396
00:22:34,540 --> 00:22:35,380
And so forth.

397
00:22:35,380 --> 00:22:43,110
These values at which b sub n is
equal to 3/4, b is equal to

398
00:22:43,110 --> 00:22:46,530
little b plus 3/4 get
more and more rare.

399
00:22:46,530 --> 00:22:51,560
So in some sense, this sequence
here where b2 up to

400
00:22:51,560 --> 00:22:53,190
b4 is zero.

401
00:22:53,190 --> 00:22:57,310
b6 up to b24 is zero
and so forth.

402
00:22:57,310 --> 00:23:00,580
This is some kind of
convergence, also.

403
00:23:00,580 --> 00:23:05,340
But it's not what anyone
would call convergence.

404
00:23:05,340 --> 00:23:08,610
I mean, as far as numbers are
concerned, there's only one

405
00:23:08,610 --> 00:23:11,950
kind of convergence that people
ever talk about, and

406
00:23:11,950 --> 00:23:13,930
it's this kind of convergence
here.

407
00:23:13,930 --> 00:23:17,070
This, although these numbers
are getting close

408
00:23:17,070 --> 00:23:18,820
to b in some sense.

409
00:23:18,820 --> 00:23:21,630
That's not viewed
as convergence.

410
00:23:21,630 --> 00:23:26,950
So here, even though almost all
the numbers are close to

411
00:23:26,950 --> 00:23:30,420
b, they don't stay close
to b, in a sense.

412
00:23:30,420 --> 00:23:34,290
They always pop up at some place
in the future, and that

413
00:23:34,290 --> 00:23:37,160
destroys the whole idea
of convergence.

414
00:23:37,160 --> 00:23:41,920
It destroys most theorems
about convergence.

415
00:23:41,920 --> 00:23:45,430
That's an example where you
don't have convergence.

416
00:23:45,430 --> 00:23:49,100
OK, random variables are really
a lot more complicated

417
00:23:49,100 --> 00:23:50,290
than numbers.

418
00:23:50,290 --> 00:23:57,000
I mean, a random variable is a
function from the sample space

419
00:23:57,000 --> 00:23:59,010
to real numbers.

420
00:23:59,010 --> 00:24:00,800
All of you know that's
not really what a

421
00:24:00,800 --> 00:24:02,470
random variable is.

422
00:24:02,470 --> 00:24:06,100
All of you know that a random
variable is a number that

423
00:24:06,100 --> 00:24:10,110
wiggles around a little bit,
rather than being fixed at

424
00:24:10,110 --> 00:24:12,695
what you ordinarily think of
a number as being, right?

425
00:24:16,990 --> 00:24:22,350
Since that's a very imprecise
notion, and the precise notion

426
00:24:22,350 --> 00:24:26,150
is very complicated, to build up
your intuition about this,

427
00:24:26,150 --> 00:24:29,830
you have to really think hard
about what convergence of

428
00:24:29,830 --> 00:24:32,300
random variables means.

429
00:24:32,300 --> 00:24:35,230
For convergence and
distribution, it's not the

430
00:24:35,230 --> 00:24:38,140
random variables, but the
distribution function of the

431
00:24:38,140 --> 00:24:40,100
random variables
that converge.

432
00:24:40,100 --> 00:24:43,710
In other words, in the
distribution function of z sub

433
00:24:43,710 --> 00:24:46,940
n, where you have a sequence of
random variables, z1, z2,

434
00:24:46,940 --> 00:24:50,520
z3 and so forth, the
distribution function

435
00:24:50,520 --> 00:24:56,890
evaluated at each real value z
converges for each z in the

436
00:24:56,890 --> 00:25:01,370
case where the distribution
function of this final

437
00:25:01,370 --> 00:25:04,740
convergent random variable
is continuous.

438
00:25:04,740 --> 00:25:06,600
We all studied that.

439
00:25:06,600 --> 00:25:08,370
We know what that means now.

440
00:25:08,370 --> 00:25:11,490
For convergence and probability,
we talked about

441
00:25:11,490 --> 00:25:14,270
convergence and probability
in two ways.

442
00:25:14,270 --> 00:25:16,910
One with an epsilon
and a delta.

443
00:25:16,910 --> 00:25:19,480
And then saying for every
epsilon and delta that isn't

444
00:25:19,480 --> 00:25:22,130
big enough, something happens.

445
00:25:22,130 --> 00:25:25,730
And then we saw that it was a
little easier to describe.

446
00:25:25,730 --> 00:25:28,570
It was a little easier to drive
describe by saying the

447
00:25:28,570 --> 00:25:30,880
convergence in probability.

448
00:25:30,880 --> 00:25:33,460
These distribution
functions have to

449
00:25:33,460 --> 00:25:34,810
converge to a unit step.

450
00:25:34,810 --> 00:25:35,620
And that's enough.

451
00:25:35,620 --> 00:25:39,230
They converge to a unit steps
at every z except

452
00:25:39,230 --> 00:25:41,570
where the step is.

453
00:25:41,570 --> 00:25:42,920
We talked about that.

454
00:25:42,920 --> 00:25:46,130
For convergence with probability
one, and this is

455
00:25:46,130 --> 00:25:50,030
the thing we want to talk about
today, this is the one

456
00:25:50,030 --> 00:25:54,020
that sounds so easy, and
which is really tricky.

457
00:25:54,020 --> 00:25:55,990
I don't want to scare
you about this.

458
00:25:55,990 --> 00:26:00,660
If you're not scared about
it to start with, I don't

459
00:26:00,660 --> 00:26:01,450
want to scare you.

460
00:26:01,450 --> 00:26:05,670
But I would like to convince
you that if you think you

461
00:26:05,670 --> 00:26:09,260
understand it and you haven't
spent a lot of time thinking

462
00:26:09,260 --> 00:26:12,070
about it, you're probably
due for a rude

463
00:26:12,070 --> 00:26:14,370
awakening at some point.

464
00:26:14,370 --> 00:26:17,870
So for convergence with
probability one, the set of

465
00:26:17,870 --> 00:26:22,370
sample paths that converge
has probability one.

466
00:26:22,370 --> 00:26:27,930
In other words, the sequence Y1,
Y2 converges to zero with

467
00:26:27,930 --> 00:26:30,290
probability one.

468
00:26:30,290 --> 00:26:33,070
And now I'm going to talk
about converging to zero

469
00:26:33,070 --> 00:26:35,820
rather than converging to
some random variable.

470
00:26:35,820 --> 00:26:39,120
Because if you're interested
in a sequence of random

471
00:26:39,120 --> 00:26:43,550
variables Z1, Z2 that converge
to some other random variable

472
00:26:43,550 --> 00:26:47,570
Z, you can get rid of a lot of
the complication by just

473
00:26:47,570 --> 00:26:51,310
saying, let's define a random
variable y sub n, which is

474
00:26:51,310 --> 00:26:53,570
equal to z sub n minus c.

475
00:26:53,570 --> 00:26:56,430
And then what we're interested
in is do these random

476
00:26:56,430 --> 00:26:59,490
variables y sub n
converged to 0.

477
00:26:59,490 --> 00:27:02,980
We can forget about what it's
converging to, and only worry

478
00:27:02,980 --> 00:27:05,560
about it converging to 0.

479
00:27:05,560 --> 00:27:11,270
OK, so when we do that, this
sequence of random variables

480
00:27:11,270 --> 00:27:15,160
converges to 0 with
probability 1.

481
00:27:15,160 --> 00:27:22,800
If the probability of the set of
sample points for which the

482
00:27:22,800 --> 00:27:25,410
sample path converges to 0.

483
00:27:25,410 --> 00:27:31,310
If that set of sample paths
has probability 1--

484
00:27:31,310 --> 00:27:36,050
namely, for almost everything
in the space, for almost

485
00:27:36,050 --> 00:27:39,820
everything in its peculiar
sense of probability--

486
00:27:39,820 --> 00:27:44,050
if that holds true, then you say
you have convergence with

487
00:27:44,050 --> 00:27:45,300
probability 1.

488
00:27:49,440 --> 00:27:51,440
Now, that looks straightforward,

489
00:27:51,440 --> 00:27:54,770
and I hope it is.

490
00:27:54,770 --> 00:27:56,590
You can memorize it
or do whatever you

491
00:27:56,590 --> 00:27:59,270
want to do with it.

492
00:27:59,270 --> 00:28:02,340
We're going to go on now and
prove an important theorem

493
00:28:02,340 --> 00:28:05,260
about convergence with
probability 1.

494
00:28:05,260 --> 00:28:09,160
I'm going to give a proof here
in class that's a little more

495
00:28:09,160 --> 00:28:12,610
detailed than the proof
I give in the notes.

496
00:28:12,610 --> 00:28:15,280
I don't like to give
proofs in class.

497
00:28:15,280 --> 00:28:19,275
I think it's a lousy idea
because when you're studying a

498
00:28:19,275 --> 00:28:22,940
proof, you have to go
at your own pace.

499
00:28:22,940 --> 00:28:29,420
But the problem is, I
know that students--

500
00:28:29,420 --> 00:28:31,200
and I was once a
student myself,

501
00:28:31,200 --> 00:28:33,170
and I'm still a student.

502
00:28:33,170 --> 00:28:38,165
If I see a proof, I will only
look at enough of it to say,

503
00:28:38,165 --> 00:28:40,000
ah, I get the idea of it.

504
00:28:40,000 --> 00:28:41,610
And then I will stop.

505
00:28:41,610 --> 00:28:44,840
And for this one, you need a
little more than the idea of

506
00:28:44,840 --> 00:28:46,410
it because it's something
we're going to

507
00:28:46,410 --> 00:28:48,020
build on all the time.

508
00:28:48,020 --> 00:28:51,300
So I want to go through
this proof carefully.

509
00:28:51,300 --> 00:28:56,990
And I hope that most of you
will follow most of it.

510
00:28:56,990 --> 00:28:59,560
And the parts of it that you
don't follow, I hope you'll go

511
00:28:59,560 --> 00:29:02,010
back and think about
it, because

512
00:29:02,010 --> 00:29:05,290
this is really important.

513
00:29:05,290 --> 00:29:10,630
OK, so the theorem says, let
this sequence of random

514
00:29:10,630 --> 00:29:17,080
variables satisfy the expected
value of the magnitude of Y

515
00:29:17,080 --> 00:29:23,360
sub n, the sum from n equals 1
to infinity of this is less

516
00:29:23,360 --> 00:29:24,610
than infinity.

517
00:29:27,740 --> 00:29:29,990
As usual there's a
misprint there.

518
00:29:29,990 --> 00:29:33,280
The sum, the expected
value of Yn, the

519
00:29:33,280 --> 00:29:34,970
bracket should be there.

520
00:29:34,970 --> 00:29:36,560
It's supposed to be less
than infinity.

521
00:29:40,820 --> 00:29:42,780
Let me write that down.

522
00:29:42,780 --> 00:29:49,720
The sum from n equals 1 to
infinity of expected value of

523
00:29:49,720 --> 00:29:56,330
the magnitude of Y sub n
is less than infinity.

524
00:29:56,330 --> 00:29:59,340
So it's a finite sum.

525
00:29:59,340 --> 00:30:03,960
So we're talking about these
Yn's when we start talking

526
00:30:03,960 --> 00:30:07,350
about the strong law
of large numbers.

527
00:30:07,350 --> 00:30:12,300
Yn is going to be something like
the sum from n equals 1

528
00:30:12,300 --> 00:30:15,730
to m divided by m.

529
00:30:15,730 --> 00:30:17,800
In other words, it's going to
be the sample average, or

530
00:30:17,800 --> 00:30:19,090
something like that.

531
00:30:19,090 --> 00:30:22,650
And these sample averages, if
you have a mean 0, are going

532
00:30:22,650 --> 00:30:23,850
to get small.

533
00:30:23,850 --> 00:30:26,620
The question is, when you sum
all of these things that are

534
00:30:26,620 --> 00:30:31,120
getting small, do you still get
something which is small?

535
00:30:31,120 --> 00:30:34,490
When you're dealing with the
weak law of large numbers,

536
00:30:34,490 --> 00:30:37,640
it's not necessary that
that sum gets small.

537
00:30:37,640 --> 00:30:40,910
It's only necessary that each
of the terms get small.

538
00:30:40,910 --> 00:30:43,660
Here we're saying, let's
assume also that

539
00:30:43,660 --> 00:30:46,560
this sum gets small.

540
00:30:46,560 --> 00:30:50,930
OK, so we want to prove that
under this condition, all of

541
00:30:50,930 --> 00:30:57,430
these sequences with probability
1 converge to 0,

542
00:30:57,430 --> 00:31:00,690
the individual sequences
converge.

543
00:31:00,690 --> 00:31:04,520
OK, so let's go through
the proof now.

544
00:31:04,520 --> 00:31:08,560
And as I say, I won't do
this to you very often.

545
00:31:08,560 --> 00:31:14,110
But I think for this one,
it's sort of necessary.

546
00:31:14,110 --> 00:31:18,000
OK, so first we'll use the
Markov inequality.

547
00:31:18,000 --> 00:31:22,310
And I'm dealing with a finite
value of m here.

548
00:31:22,310 --> 00:31:26,900
The probability that the sum of
a finite set of these Y sub

549
00:31:26,900 --> 00:31:30,960
n's is greater than alpha is
less than or equal to the

550
00:31:30,960 --> 00:31:33,890
expected value of that
random variable.

551
00:31:33,890 --> 00:31:36,730
Namely, this random
variable here.

552
00:31:36,730 --> 00:31:40,930
Sum from n equals 1 to m of
magnitude of Y sub n.

553
00:31:40,930 --> 00:31:43,560
That's just a random variable.

554
00:31:43,560 --> 00:31:46,640
And the probability that that
random variable is greater

555
00:31:46,640 --> 00:31:49,610
than alpha is less than or equal
to the expected value of

556
00:31:49,610 --> 00:31:52,980
that random variable
divided by alpha.

557
00:31:52,980 --> 00:32:02,850
OK, well now, this quantity here
is increasing in Y sub n.

558
00:32:02,850 --> 00:32:06,510
The magnitude of Y sub n is
a non-negative quantity.

559
00:32:06,510 --> 00:32:10,860
You take the expectation of a
non-negative quantity, if it

560
00:32:10,860 --> 00:32:15,470
has an expectation, which we're
assuming here for this

561
00:32:15,470 --> 00:32:18,680
to be less infinity, all of
these things have to have

562
00:32:18,680 --> 00:32:20,800
expectations.

563
00:32:20,800 --> 00:32:26,990
So as we increase m, this
gets bigger and bigger.

564
00:32:26,990 --> 00:32:31,720
So this quantity here is going
to be less than or equal to

565
00:32:31,720 --> 00:32:35,370
the sum from n equals 1 to
infinity of expected value of

566
00:32:35,370 --> 00:32:37,470
Y sub n divided by alpha.

567
00:32:37,470 --> 00:32:41,830
What I'm being careful about
here is all of the things that

568
00:32:41,830 --> 00:32:47,170
happen when you go from finite
m to infinite m.

569
00:32:47,170 --> 00:32:52,260
And I'm using what you know
about finite m, and then being

570
00:32:52,260 --> 00:32:56,080
very careful about going
to infinite m.

571
00:32:56,080 --> 00:32:59,340
And I'm going to try to explain
why as we do it.

572
00:32:59,340 --> 00:33:01,570
But here, it's straightforward.

573
00:33:01,570 --> 00:33:07,090
The expected value of a finite
sum is equal to the finite sum

574
00:33:07,090 --> 00:33:08,580
of an expected value.

575
00:33:08,580 --> 00:33:11,560
When you go to the limit, m goes
to infinity, you don't

576
00:33:11,560 --> 00:33:15,230
know whether these expected
values exist or not.

577
00:33:15,230 --> 00:33:18,990
You're sort of confused on both
sides of this equation.

578
00:33:18,990 --> 00:33:21,550
So we're sticking to
finite values here.

579
00:33:21,550 --> 00:33:25,710
Then, we're taking this
quantity, going to the limit

580
00:33:25,710 --> 00:33:27,450
as m goes to infinity.

581
00:33:27,450 --> 00:33:30,570
This quantity has to get bigger
and bigger as m goes to

582
00:33:30,570 --> 00:33:33,870
infinity, so this quantity
has to be less

583
00:33:33,870 --> 00:33:36,070
than or equal to this.

584
00:33:36,070 --> 00:33:40,980
This now, for a given alpha,
is just a number.

585
00:33:40,980 --> 00:33:43,820
It's nothing more than a number,
so we can deal with

586
00:33:43,820 --> 00:33:47,420
this pretty easily as we
make alpha big enough.

587
00:33:47,420 --> 00:33:50,050
But for most of the argument,
we're going to view alpha as

588
00:33:50,050 --> 00:33:51,700
being fixed.

589
00:33:51,700 --> 00:33:58,590
OK, so now the probability that
this sum, finite sum is

590
00:33:58,590 --> 00:34:01,920
greater than alpha, is less
than or equal to this.

591
00:34:01,920 --> 00:34:03,620
This was the thing we just
finished proving

592
00:34:03,620 --> 00:34:04,870
on the other page.

593
00:34:09,110 --> 00:34:11,600
This is less than or
equal to that.

594
00:34:11,600 --> 00:34:17,969
That's what I repeated, so I'm
not cheating you at all here.

595
00:34:17,969 --> 00:34:21,400
Now, it's a pain to write
that down all the time.

596
00:34:21,400 --> 00:34:27,580
So let's let the set, A sub m,
be the set of sample points

597
00:34:27,580 --> 00:34:32,380
such that its finite sum
of Y sub n of omega is

598
00:34:32,380 --> 00:34:34,170
greater than alpha.

599
00:34:34,170 --> 00:34:35,420
This is a random--

600
00:34:39,260 --> 00:34:43,150
for each value of omega,
this is just a number.

601
00:34:43,150 --> 00:34:49,710
The sum of the magnitude of Y
sub n is a random variable.

602
00:34:49,710 --> 00:34:53,270
It takes on a numerical
value for every omega

603
00:34:53,270 --> 00:34:54,760
in the sample space.

604
00:34:54,760 --> 00:34:59,140
So A sub m is the set of points
in the sample space for

605
00:34:59,140 --> 00:35:02,600
which this quantity here
is bigger than alpha.

606
00:35:02,600 --> 00:35:08,010
So we can rewrite this
now as just the

607
00:35:08,010 --> 00:35:10,440
probability of A sub m.

608
00:35:10,440 --> 00:35:14,680
So this is equivalent to saying,
the probability of A

609
00:35:14,680 --> 00:35:18,120
sub m is less than or equal
to this number here.

610
00:35:18,120 --> 00:35:20,700
For a fixed alpha,
this is a number.

611
00:35:20,700 --> 00:35:24,500
This is something which
can vary with m.

612
00:35:24,500 --> 00:35:29,560
Since these numbers here now,
now we're dealing with a

613
00:35:29,560 --> 00:35:32,710
sample space, which is
a little strange.

614
00:35:32,710 --> 00:35:39,570
We're talking about sample
points and we're saying, this

615
00:35:39,570 --> 00:35:46,320
number here, this magnitude of Y
sub n at a particular sample

616
00:35:46,320 --> 00:35:50,040
point omega is greater
than or equal to 0.

617
00:35:50,040 --> 00:35:55,050
Therefore, a sub m is a subset
of A sub m plus 1.

618
00:35:55,050 --> 00:36:03,650
In other words, as m gets larger
and larger here, m here

619
00:36:03,650 --> 00:36:05,450
gets larger and larger.

620
00:36:05,450 --> 00:36:08,950
Therefore, this sum here
gets larger and larger.

621
00:36:08,950 --> 00:36:14,240
Therefore, the set of omega for
which this increasing sum

622
00:36:14,240 --> 00:36:17,370
is greater than alpha gets
bigger and bigger.

623
00:36:17,370 --> 00:36:20,430
And that's the thing that we're
saying here, A sub m is

624
00:36:20,430 --> 00:36:24,990
included in A sub m plus 1 for
m greater than or equal to 1.

625
00:36:24,990 --> 00:36:28,920
OK, so the left side of this
quantity here, as a function

626
00:36:28,920 --> 00:36:32,370
of m, is a non-decreasing
bounded

627
00:36:32,370 --> 00:36:34,305
sequence of real numbers.

628
00:36:41,420 --> 00:36:43,090
Yes, the probability
of something

629
00:36:43,090 --> 00:36:44,120
is just a real number.

630
00:36:44,120 --> 00:36:46,970
A probability is a number.

631
00:36:46,970 --> 00:36:50,230
So this quantity here
is a real number.

632
00:36:50,230 --> 00:36:53,810
It's a real number which is
non-decreasing, so it keeps

633
00:36:53,810 --> 00:36:54,920
moving upward.

634
00:36:54,920 --> 00:36:59,330
What I'm trying to do now
is now, I went to

635
00:36:59,330 --> 00:37:00,300
the limit over here.

636
00:37:00,300 --> 00:37:03,110
I want to go to the
limit here.

637
00:37:03,110 --> 00:37:08,680
And so I have a sequence
of numbers in m.

638
00:37:08,680 --> 00:37:12,500
This sequence of numbers
is non-decreasing.

639
00:37:12,500 --> 00:37:14,250
So it's moving up.

640
00:37:14,250 --> 00:37:17,110
Every one of those quantities
is bounded by

641
00:37:17,110 --> 00:37:18,840
this quantity here.

642
00:37:18,840 --> 00:37:23,350
So I have an increasing sequence
of real numbers,

643
00:37:23,350 --> 00:37:25,960
which is bounded on the top.

644
00:37:25,960 --> 00:37:28,280
What happens?

645
00:37:28,280 --> 00:37:30,810
When you have a sequence
of real

646
00:37:30,810 --> 00:37:32,780
numbers which is bounded--

647
00:37:36,200 --> 00:37:38,810
I have a slide to prove this,
but I'm not going to prove it

648
00:37:38,810 --> 00:37:40,953
because it's tedious.

649
00:37:46,470 --> 00:37:53,210
Here we have this probability
which I'm calling A sub m

650
00:37:53,210 --> 00:37:59,740
probability of A sub m.

651
00:37:59,740 --> 00:38:07,730
Here I have the probability of
A sub m plus 1, and so forth.

652
00:38:07,730 --> 00:38:09,980
Here I have this
limit up here.

653
00:38:09,980 --> 00:38:13,850
All of this sequence of numbers,
there's an infinite

654
00:38:13,850 --> 00:38:15,770
sequence of them.

655
00:38:15,770 --> 00:38:17,650
They're all non-decreasing.

656
00:38:17,650 --> 00:38:20,780
They're all bounded by
this number here.

657
00:38:20,780 --> 00:38:23,300
And what happens?

658
00:38:23,300 --> 00:38:27,600
Well, either we go up to there
as a limit or else we stop

659
00:38:27,600 --> 00:38:30,890
sometime earlier as a limit.

660
00:38:30,890 --> 00:38:36,600
I should prove this, but it's
something we use all the time.

661
00:38:36,600 --> 00:38:43,190
It's a sequence of increasing
or non-decreasing numbers.

662
00:38:43,190 --> 00:38:46,420
If it's bounded by something, it
has to have a finite limit.

663
00:38:46,420 --> 00:38:49,390
The limit is less than or
equal to this quantity.

664
00:38:49,390 --> 00:38:53,690
It might be strictly less, but
the limit has to exist.

665
00:38:53,690 --> 00:38:55,620
And the limit has to be less
than or equal to b.

666
00:38:59,900 --> 00:39:02,240
OK, that's what we're
saying here.

667
00:39:02,240 --> 00:39:10,510
When we go to this limit, this
limit of the probability of A

668
00:39:10,510 --> 00:39:14,380
sub m is less than or equal
to this number here.

669
00:39:14,380 --> 00:39:19,310
OK, if I use this property of
nesting intervals, when you

670
00:39:19,310 --> 00:39:26,860
have A sub 1 nested inside of
A sub 2, nested inside of A

671
00:39:26,860 --> 00:39:33,380
sub 3, what we'd like to go
do is go to this limit.

672
00:39:33,380 --> 00:39:35,210
The limit, unfortunately,
doesn't make

673
00:39:35,210 --> 00:39:36,460
any sense in general.

674
00:39:39,130 --> 00:39:43,240
With this property of the
axioms, it's equation number 9

675
00:39:43,240 --> 00:39:47,490
in chapter 1 says that
we can do something

676
00:39:47,490 --> 00:39:51,190
that's almost as good.

677
00:39:51,190 --> 00:40:00,630
What it says that as we go to
this limit here, what we get

678
00:40:00,630 --> 00:40:06,830
is that this limit is
the probability of

679
00:40:06,830 --> 00:40:08,960
this infinite union.

680
00:40:08,960 --> 00:40:13,410
That's equal to the limit
as m goes to infinity of

681
00:40:13,410 --> 00:40:16,655
probability of A sub m.

682
00:40:16,655 --> 00:40:20,810
OK, look up equation 9,
and you'll see that's

683
00:40:20,810 --> 00:40:22,120
exactly what it says.

684
00:40:22,120 --> 00:40:25,870
If you think this is
obvious, it's not.

685
00:40:25,870 --> 00:40:29,350
It ain't obvious at all because
it's not even clear

686
00:40:29,350 --> 00:40:32,380
that this--

687
00:40:32,380 --> 00:40:35,090
well, nothing very much about
this union is clear.

688
00:40:35,090 --> 00:40:38,570
We know that this union must
be a measurable set.

689
00:40:38,570 --> 00:40:40,280
It must have a probability.

690
00:40:40,280 --> 00:40:42,090
We don't know much
more about it.

691
00:40:42,090 --> 00:40:45,340
But anyway, that property tells
us that this is true.

692
00:41:03,190 --> 00:41:06,040
OK, so where we are
at this point.

693
00:41:06,040 --> 00:41:09,680
I don't think I've skip
something, have I?

694
00:41:09,680 --> 00:41:13,430
Oh, no, that's the thing I
didn't want to talk about.

695
00:41:13,430 --> 00:41:19,720
OK, so A sub m is a set
of omega which satisfy

696
00:41:19,720 --> 00:41:21,970
this for finite m.

697
00:41:21,970 --> 00:41:28,230
The probability of this union
is then the union of all of

698
00:41:28,230 --> 00:41:30,470
these quantities over all m.

699
00:41:30,470 --> 00:41:35,050
And this is less than or equal
to this bound that we had.

700
00:41:35,050 --> 00:41:44,890
OK, so I even hate giving proofs
of this sort because

701
00:41:44,890 --> 00:41:46,960
it's a set of simple ideas.

702
00:41:46,960 --> 00:41:49,980
To track down every one
of them is difficult.

703
00:41:49,980 --> 00:41:53,320
The text doesn't track down
every one of them.

704
00:41:53,320 --> 00:41:55,750
And that's what I'm
trying to do here.

705
00:42:06,580 --> 00:42:11,870
We have two possibilities here,
and we're looking at

706
00:42:11,870 --> 00:42:13,330
this limit here.

707
00:42:13,330 --> 00:42:20,390
This limiting sum, which for
each omega is just a sequence,

708
00:42:20,390 --> 00:42:23,500
a non-decreasing sequence
of real numbers.

709
00:42:23,500 --> 00:42:27,940
So one possibility is that this
sequence of real numbers

710
00:42:27,940 --> 00:42:30,000
is bigger than alpha.

711
00:42:30,000 --> 00:42:32,390
The other possibility
is that it's less

712
00:42:32,390 --> 00:42:34,050
than or equal to alpha.

713
00:42:34,050 --> 00:42:37,660
If it's less than or equal to
alpha, then every one of these

714
00:42:37,660 --> 00:42:43,920
numbers is less than or equal
to alpha and omega has to be

715
00:42:43,920 --> 00:42:47,230
not in this union here.

716
00:42:47,230 --> 00:42:50,800
If the sum is bigger than
alpha, then one of the

717
00:42:50,800 --> 00:42:53,500
elements in this set is
bigger than alpha and

718
00:42:53,500 --> 00:42:55,550
omega is in this set.

719
00:42:55,550 --> 00:43:00,300
So what all of that says, and
you're just going to have to

720
00:43:00,300 --> 00:43:03,301
look at that because
it's not--

721
00:43:03,301 --> 00:43:05,760
it's one of these tedious
arguments.

722
00:43:05,760 --> 00:43:09,690
So the probability of omega such
that this sum is greater

723
00:43:09,690 --> 00:43:13,030
than alpha is less than or equal
to this number here.

724
00:43:13,030 --> 00:43:16,770
At this point, we have
made a major change

725
00:43:16,770 --> 00:43:19,000
in what we're doing.

726
00:43:19,000 --> 00:43:24,750
Before we were talking about
numbers like probabilities,

727
00:43:24,750 --> 00:43:27,400
numbers like expected values.

728
00:43:27,400 --> 00:43:33,020
Here, suddenly, we are talking
about sample points.

729
00:43:33,020 --> 00:43:35,870
We're talking about the
probability of a set of sample

730
00:43:35,870 --> 00:43:40,340
points, such that the sum
is greater than alpha.

731
00:43:40,340 --> 00:43:41,358
Yes?

732
00:43:41,358 --> 00:43:43,207
AUDIENCE: I understand how if
the whole sum if less than or

733
00:43:43,207 --> 00:43:45,840
equal than alpha then
every element is.

734
00:43:45,840 --> 00:43:49,160
But did you say that if it's
greater than alpha, then at

735
00:43:49,160 --> 00:43:50,360
least one element is
greater than alpha?

736
00:43:50,360 --> 00:43:51,610
Why is that?

737
00:43:57,660 --> 00:44:00,430
PROFESSOR: Well, because either
the sum is less than or

738
00:44:00,430 --> 00:44:03,670
equal to alpha or it's
greater than alpha.

739
00:44:03,670 --> 00:44:09,230
And if it's less than or equal
to alpha, then omega

740
00:44:09,230 --> 00:44:11,570
is not in this set.

741
00:44:11,570 --> 00:44:17,860
So the alternative is that omega
has to be in this set.

742
00:44:17,860 --> 00:44:20,540
Except the other way of looking
at it is if you have a

743
00:44:20,540 --> 00:44:25,390
sequence of numbers, which is
approaching a limit, and that

744
00:44:25,390 --> 00:44:31,870
limit is bigger than alpha, then
one of the terms has to

745
00:44:31,870 --> 00:44:33,990
be bigger than alpha.

746
00:44:33,990 --> 00:44:34,924
Yes?

747
00:44:34,924 --> 00:44:36,406
AUDIENCE: I think the confusion
is between the

748
00:44:36,406 --> 00:44:38,876
partial sums and the
terms of the sum.

749
00:44:38,876 --> 00:44:40,852
That's what he's confusing.

750
00:44:40,852 --> 00:44:43,322
Does that make sense?

751
00:44:43,322 --> 00:44:45,298
He's saying instead of
each partial sum, not

752
00:44:45,298 --> 00:44:46,548
each term in the sum.

753
00:44:50,750 --> 00:44:52,000
PROFESSOR: Yes.

754
00:44:58,540 --> 00:45:00,510
Except I don't see how that
answers the question.

755
00:45:11,000 --> 00:45:16,630
Except the point here is, if
each partial sum is less than

756
00:45:16,630 --> 00:45:20,260
or equal to alpha, then the
limit has to be less than or

757
00:45:20,260 --> 00:45:20,960
equal to alpha.

758
00:45:20,960 --> 00:45:22,450
That's what I was saying
on the other page.

759
00:45:22,450 --> 00:45:25,130
If you have a sequence of
numbers, which has an upper

760
00:45:25,130 --> 00:45:28,810
bound on them, then you
have to have a limit.

761
00:45:28,810 --> 00:45:31,900
And that limit has to be less
than or equal to alpha.

762
00:45:31,900 --> 00:45:34,090
So that's this case here.

763
00:45:34,090 --> 00:45:38,660
We have a sum of numbers as
we're going to the limit as m

764
00:45:38,660 --> 00:45:42,490
gets larger and larger,
these partial sums

765
00:45:42,490 --> 00:45:45,210
have to go to a limit.

766
00:45:45,210 --> 00:45:48,070
The partial sums are all less
than or equal to alpha.

767
00:45:48,070 --> 00:45:51,130
Then the infinite sum is less
than or equal to alpha, and

768
00:45:51,130 --> 00:45:53,810
omega is not in this set here.

769
00:45:53,810 --> 00:45:55,540
And otherwise, it is.

770
00:45:55,540 --> 00:45:59,220
OK, if I talk more about it,
I'll get more confused.

771
00:45:59,220 --> 00:46:01,985
So I think the slides
are clear.

772
00:46:09,020 --> 00:46:17,710
Now, if we look at the case
where alpha is greater than or

773
00:46:17,710 --> 00:46:26,180
equal to this sum, and we take
the complement of the set, the

774
00:46:26,180 --> 00:46:30,380
probability of the set of omega
for which this sum is

775
00:46:30,380 --> 00:46:32,900
less than or equal
to alpha has--

776
00:46:32,900 --> 00:46:36,830
oh, let's forget about
this for the moment.

777
00:46:36,830 --> 00:46:40,440
If I take the complement of this
set, the probability of

778
00:46:40,440 --> 00:46:43,560
the set of omega, such that the
sum is less than or equal

779
00:46:43,560 --> 00:46:47,190
to alpha, is greater
than 1 minus this

780
00:46:47,190 --> 00:46:48,820
expected value here.

781
00:46:48,820 --> 00:46:51,480
Now I'm saying, let's look at
the case where alpha is big

782
00:46:51,480 --> 00:46:55,280
enough that it's greater
than this number here.

783
00:46:55,280 --> 00:47:00,820
So this probability is greater
than 1 minus this number.

784
00:47:00,820 --> 00:47:06,720
So if the sum is less than or
equal to alpha for any given

785
00:47:06,720 --> 00:47:12,220
omega, then this quantity
here converges.

786
00:47:12,220 --> 00:47:14,620
Now I'm talking about
sample sequences.

787
00:47:14,620 --> 00:47:18,630
I'm saying I have an increasing
sequence of numbers

788
00:47:18,630 --> 00:47:21,540
corresponding to one particular
sample point.

789
00:47:21,540 --> 00:47:25,500
This increasing set of numbers
is less than or equal.

790
00:47:25,500 --> 00:47:28,760
Each element of it is less than
or equal to alpha, so the

791
00:47:28,760 --> 00:47:31,820
limit of it is less than
or equal to alpha.

792
00:47:31,820 --> 00:47:37,050
And what that says is the limit
of Y sub n of omega,

793
00:47:37,050 --> 00:47:39,760
this has to be equal to 0
for that sample point.

794
00:47:39,760 --> 00:47:41,500
This is all the sample
point argument.

795
00:47:46,280 --> 00:47:53,500
And what that says then is the
probability of omega, such

796
00:47:53,500 --> 00:47:57,580
that this limit here is equal
to 0, that's this quantity

797
00:47:57,580 --> 00:48:01,310
here, which is the same as this
quantity, which has to be

798
00:48:01,310 --> 00:48:02,935
greater than this quantity.

799
00:48:08,570 --> 00:48:11,100
This implies this.

800
00:48:11,100 --> 00:48:16,260
Therefore, the probability of
this has to be bigger than

801
00:48:16,260 --> 00:48:17,950
this probability here.

802
00:48:17,950 --> 00:48:22,480
Now, if we let alpha go to
infinity, what that says is

803
00:48:22,480 --> 00:48:26,310
this quantity goes to 0 and the
probability of the set of

804
00:48:26,310 --> 00:48:30,730
omega, such that this limit is
equal to 0, is equal to 1.

805
00:48:35,010 --> 00:48:39,640
I think if I try to spend 20
more minutes talking about

806
00:48:39,640 --> 00:48:42,860
that in more detail, it
won't get any clearer.

807
00:48:42,860 --> 00:48:47,020
It is one of these very tedious
arguments where you

808
00:48:47,020 --> 00:48:51,380
have to sit down and follow
it step by step.

809
00:48:51,380 --> 00:48:54,500
I wrote the steps that's
very carefully.

810
00:48:54,500 --> 00:49:02,180
And at this point, I have
to leave it as it is.

811
00:49:02,180 --> 00:49:05,540
But the theorem has been proven,
at least in what's

812
00:49:05,540 --> 00:49:08,800
written, if not in
what I've said.

813
00:49:08,800 --> 00:49:13,050
OK, let's look at an example
of this now.

814
00:49:13,050 --> 00:49:17,110
Let's look at the example where
these random variables Y

815
00:49:17,110 --> 00:49:21,930
sub n for n greater than or
equal to 1, have this

816
00:49:21,930 --> 00:49:23,170
following property.

817
00:49:23,170 --> 00:49:26,190
It's almost the same as the
sequence of numbers I talked

818
00:49:26,190 --> 00:49:27,830
about before.

819
00:49:27,830 --> 00:49:31,990
But what I'm going
to do now is--

820
00:49:31,990 --> 00:49:34,630
these are not IID random
variables.

821
00:49:34,630 --> 00:49:39,130
If they're IID random variables,
you're never going

822
00:49:39,130 --> 00:49:41,310
to talk about the sum
being finite.

823
00:49:41,310 --> 00:49:44,260
Sum of the expected values
being finite.

824
00:49:44,260 --> 00:49:52,620
How they behave is that for one
less than or equal to 5,

825
00:49:52,620 --> 00:49:55,520
you pick one of these random
variables in here and make it

826
00:49:55,520 --> 00:49:56,690
equal to 1.

827
00:49:56,690 --> 00:49:59,040
And all the rest
are equal to 0.

828
00:49:59,040 --> 00:50:03,560
From 5 to 25, you pick one of
the random variables, make it

829
00:50:03,560 --> 00:50:06,400
equal to 1, and all the
others are equal to 0.

830
00:50:06,400 --> 00:50:08,130
You choose randomly in here.

831
00:50:08,130 --> 00:50:12,380
From 25 to 125, you pick
one random variable,

832
00:50:12,380 --> 00:50:14,020
set it equal to 1.

833
00:50:14,020 --> 00:50:16,910
All the other random variables,
set it equal to 0,

834
00:50:16,910 --> 00:50:19,500
and so forth forever after.

835
00:50:19,500 --> 00:50:23,600
OK, so what does that say
for the sample points?

836
00:50:23,600 --> 00:50:29,080
If I look at any particular
sample point, what I find is

837
00:50:29,080 --> 00:50:36,610
that there's one occurrence of a
sample value equal to 1 from

838
00:50:36,610 --> 00:50:38,340
here to here.

839
00:50:38,340 --> 00:50:42,370
There's exactly one that's equal
to 1 from here to here.

840
00:50:42,370 --> 00:50:45,970
There's exactly one that's equal
to 1 from here to way

841
00:50:45,970 --> 00:50:49,890
out here at 125, and so forth.

842
00:50:49,890 --> 00:50:55,480
This is not a sequence of sample
values which converges

843
00:50:55,480 --> 00:51:00,790
because it keeps popping up
to 1 at all these values.

844
00:51:00,790 --> 00:51:05,300
So for every omega, Yn
of omega is 1 for

845
00:51:05,300 --> 00:51:07,530
some n in this interval.

846
00:51:07,530 --> 00:51:10,110
For every j and it's
0 elsewhere.

847
00:51:10,110 --> 00:51:15,830
This Yn of omega doesn't
converge for omega.

848
00:51:15,830 --> 00:51:19,310
So the probability that that
sequence converges

849
00:51:19,310 --> 00:51:22,390
is not 1, it's 0.

850
00:51:22,390 --> 00:51:26,080
So this is a particularly
awful example.

851
00:51:26,080 --> 00:51:29,490
This is a sequence of random
variables, which does not

852
00:51:29,490 --> 00:51:31,970
converge with probability 1.

853
00:51:31,970 --> 00:51:38,440
At the same time, the expected
value of Y sub n is 1 over 5

854
00:51:38,440 --> 00:51:42,320
to the j plus 1 minus
5 to the j.

855
00:51:42,320 --> 00:51:46,680
That's the probability that you
pick that particular n for

856
00:51:46,680 --> 00:51:50,310
a random variable to
be equal to 1.

857
00:51:50,310 --> 00:51:54,130
It's equal to this for 5 to the
j less than or equal to n,

858
00:51:54,130 --> 00:51:57,350
less than 5 to the j plus 1.

859
00:51:57,350 --> 00:52:01,000
When you add up all of these
things, when you add up

860
00:52:01,000 --> 00:52:04,330
expected value of Yn equal
to that over this

861
00:52:04,330 --> 00:52:06,120
interval, you get 1.

862
00:52:06,120 --> 00:52:09,460
When you add it up over the next
interval, which is much,

863
00:52:09,460 --> 00:52:11,400
much bigger, you get 1 again.

864
00:52:11,400 --> 00:52:12,950
When you add it up
over the next

865
00:52:12,950 --> 00:52:14,830
interval, you get 1 again.

866
00:52:14,830 --> 00:52:19,925
So the expected value
of the sum--

867
00:52:19,925 --> 00:52:22,850
the sum of the expected
value of the Y sub

868
00:52:22,850 --> 00:52:26,520
n's is equal to infinity.

869
00:52:26,520 --> 00:52:34,850
And what you wind up with
then is that this

870
00:52:34,850 --> 00:52:37,195
sequence does not converge--

871
00:52:49,690 --> 00:52:52,430
This says the theorem doesn't
apply at all.

872
00:52:52,430 --> 00:52:57,510
This says that the Y sub n of
omega does not converge for

873
00:52:57,510 --> 00:52:59,330
any sample function at all.

874
00:53:03,050 --> 00:53:06,910
This says that according to the
theorem, it doesn't have

875
00:53:06,910 --> 00:53:09,200
to converge.

876
00:53:09,200 --> 00:53:11,810
I mean, when you look at an
example after working very

877
00:53:11,810 --> 00:53:17,110
hard to prove a theorem, you
would like to find that if the

878
00:53:17,110 --> 00:53:25,990
conditions of the theorem are
satisfied what the theorem

879
00:53:25,990 --> 00:53:28,120
says is satisfied also.

880
00:53:28,120 --> 00:53:31,700
Here, the conditions
are not satisfied.

881
00:53:31,700 --> 00:53:33,810
And you also don't
have convergence

882
00:53:33,810 --> 00:53:35,370
with probability 1.

883
00:53:35,370 --> 00:53:39,280
You do have convergence in
probability, however.

884
00:53:39,280 --> 00:53:42,520
So this gives you a nice example
of where you have a

885
00:53:42,520 --> 00:53:47,320
sequence of random variables
that converges in probability.

886
00:53:47,320 --> 00:53:51,710
It converges in probability
because as n gets larger and

887
00:53:51,710 --> 00:53:56,730
larger, the probability that Y
sub n is going to be equal to

888
00:53:56,730 --> 00:54:00,900
anything other than 0 gets
very, very small.

889
00:54:00,900 --> 00:54:05,450
So the limit as n goes to
infinity of the probability

890
00:54:05,450 --> 00:54:07,920
that Y sub n is greater
than epsilon--

891
00:54:07,920 --> 00:54:11,750
for any epsilon greater than 0,
this probability is equal

892
00:54:11,750 --> 00:54:13,760
to 0 for all epsilon.

893
00:54:13,760 --> 00:54:17,670
So this quantity does converge
in probability.

894
00:54:17,670 --> 00:54:20,480
It does not converge
with probability 1.

895
00:54:20,480 --> 00:54:24,420
It's the simplest example I know
of where you don't have

896
00:54:24,420 --> 00:54:30,390
convergence with probability 1
and you do have convergence in

897
00:54:30,390 --> 00:54:32,290
probability.

898
00:54:32,290 --> 00:54:43,440
How about if you're looking at a
sequence of sample averages.

899
00:54:43,440 --> 00:54:47,720
Suppose you're looking at S sub
n over n where S sub n is

900
00:54:47,720 --> 00:54:51,680
a sum of IID random variables.

901
00:54:51,680 --> 00:54:57,460
Can you find an example there
where when you have a--

902
00:55:01,290 --> 00:55:05,850
can you find an example where
this sequence S sub n over n

903
00:55:05,850 --> 00:55:10,960
does converge in probability,
but does not converge with

904
00:55:10,960 --> 00:55:12,470
probability 1?

905
00:55:12,470 --> 00:55:16,660
Unfortunately, that's
very hard to do.

906
00:55:16,660 --> 00:55:20,900
And the reason is the main
theorem, which we will never

907
00:55:20,900 --> 00:55:30,210
get around to proving here is
that if you have a random

908
00:55:30,210 --> 00:55:36,510
variable x, and the expected
value of the magnitude of x is

909
00:55:36,510 --> 00:55:40,780
finite, then the strong law
of large numbers holds.

910
00:55:40,780 --> 00:55:47,530
Also, the weak law of large
numbers holds, which says that

911
00:55:47,530 --> 00:55:50,600
you're not going to find an
example where one holds and

912
00:55:50,600 --> 00:55:53,490
the other doesn't hold.

913
00:55:53,490 --> 00:55:57,910
So you have to go to strange
things like this in order to

914
00:55:57,910 --> 00:56:00,110
get these examples that
you're looking at.

915
00:56:04,220 --> 00:56:09,510
OK, let's now go from
convergence in probability 1

916
00:56:09,510 --> 00:56:14,240
to applying this to the sequence
of random variables

917
00:56:14,240 --> 00:56:23,210
where Y sub n is now equal to
the sum of n IID random

918
00:56:23,210 --> 00:56:24,520
variable divided by n.

919
00:56:24,520 --> 00:56:29,130
Namely, it's the sample average,
and we're looking at

920
00:56:29,130 --> 00:56:32,110
the limit as n goes to infinity

921
00:56:32,110 --> 00:56:33,320
of this sample average.

922
00:56:33,320 --> 00:56:40,500
What's the probability of the
set of sample points for which

923
00:56:40,500 --> 00:56:47,580
this sample path converges
to X bar?

924
00:56:47,580 --> 00:56:54,330
And the theorem says that this
quantity is equal to 1 if the

925
00:56:54,330 --> 00:56:57,940
expected value of X is
less than infinity.

926
00:56:57,940 --> 00:57:01,540
We're not going to prove that,
but what we are going to prove

927
00:57:01,540 --> 00:57:07,360
is that if the expected value
of the fourth moment of X is

928
00:57:07,360 --> 00:57:09,730
finite, then we're going
to prove that

929
00:57:09,730 --> 00:57:12,640
this theorem is true.

930
00:57:12,640 --> 00:57:20,480
OK, when we write this from now
on, we will sometimes get

931
00:57:20,480 --> 00:57:21,810
more terse.

932
00:57:21,810 --> 00:57:27,350
And instead of writing the
probability of an omega in the

933
00:57:27,350 --> 00:57:33,240
set of sample points such that
this limit for a sample point

934
00:57:33,240 --> 00:57:36,510
is equal to X bar, this whole
thing is equal to 1.

935
00:57:36,510 --> 00:57:39,480
We can sometimes write it as
the probability that this

936
00:57:39,480 --> 00:57:43,940
limit, which is now a limit
of Sn of omega over n

937
00:57:43,940 --> 00:57:45,100
is equal to X bar.

938
00:57:45,100 --> 00:57:47,450
But that's equal to 1.

939
00:57:47,450 --> 00:57:51,430
Some people write it even more
tersely as the limit of S sub

940
00:57:51,430 --> 00:57:56,590
n over n is equal to X bar
with probability 1.

941
00:57:56,590 --> 00:58:01,570
This is a very strange statement
here because this--

942
00:58:07,640 --> 00:58:11,320
I mean, what you're saying with
this statement is not

943
00:58:11,320 --> 00:58:16,770
that this limit is equal to
X bar with probability 1.

944
00:58:16,770 --> 00:58:21,610
It's saying, with probability 1,
this limit here exists for

945
00:58:21,610 --> 00:58:25,260
a sample point, and that limit
is equal to X bar.

946
00:58:25,260 --> 00:58:28,380
The thing which makes the strong
law of large numbers

947
00:58:28,380 --> 00:58:34,520
difficult is not proving
that the limit has

948
00:58:34,520 --> 00:58:36,170
a particular value.

949
00:58:36,170 --> 00:58:39,170
If there is a limit, it's
always easy to find

950
00:58:39,170 --> 00:58:40,230
what the value is.

951
00:58:40,230 --> 00:58:43,790
The thing which is difficult is
figuring out whether it has

952
00:58:43,790 --> 00:58:44,800
a limit or not.

953
00:58:44,800 --> 00:58:51,380
So this statement is fine for
people who understand what it

954
00:58:51,380 --> 00:58:55,700
says, but it's kind of
confusing otherwise.

955
00:58:55,700 --> 00:59:00,690
Still more tersely, people talk
about it as Sn over n

956
00:59:00,690 --> 00:59:04,790
goes to limit X bar with
probability 1.

957
00:59:04,790 --> 00:59:07,940
This is probably an even better
way to say it than this

958
00:59:07,940 --> 00:59:09,580
is because this is--

959
00:59:09,580 --> 00:59:12,580
I mean, this says that there's
something strange

960
00:59:12,580 --> 00:59:14,440
in the limit here.

961
00:59:14,440 --> 00:59:19,210
But I would suggest that you
write it this way until you

962
00:59:19,210 --> 00:59:20,720
get used to what it's saying.

963
00:59:20,720 --> 00:59:24,620
Because then, when you write it
this way, you realize that

964
00:59:24,620 --> 00:59:28,480
what you're talking about is
the limit over individual

965
00:59:28,480 --> 00:59:33,520
sample points rather than some
kind of more general limit.

966
00:59:33,520 --> 00:59:38,630
And convergence with probability
1 is always that

967
00:59:38,630 --> 00:59:42,420
sort of convergence.

968
00:59:42,420 --> 00:59:46,170
OK, this strong law and the
idea of convergence with

969
00:59:46,170 --> 00:59:49,240
probability 1 is really pretty
different from the other forms

970
00:59:49,240 --> 00:59:50,610
of convergence.

971
00:59:50,610 --> 00:59:54,420
In the sense that it focuses
directly on sample paths.

972
00:59:54,420 --> 00:59:59,080
The other forms of convergence
focus on things like the

973
00:59:59,080 --> 01:00:02,900
sequence of expected values,
or where the sequence of

974
01:00:02,900 --> 01:00:06,980
probabilities, or sequences of
numbers, which are the things

975
01:00:06,980 --> 01:00:09,840
you're used to dealing with.

976
01:00:09,840 --> 01:00:15,820
Here you're dealing directly
with sample points, and it

977
01:00:15,820 --> 01:00:18,740
makes it more difficult to
talk about the rate of

978
01:00:18,740 --> 01:00:21,070
convergence as n approaches
infinity.

979
01:00:21,070 --> 01:00:24,180
You can't talk about the rate
of convergence here as n

980
01:00:24,180 --> 01:00:25,860
approaches infinity.

981
01:00:25,860 --> 01:00:28,720
If you have any n less than
infinity, if you're only

982
01:00:28,720 --> 01:00:34,280
looking at a finite sequence,
you have no way of saying

983
01:00:34,280 --> 01:00:38,210
whether any of the sample values
over that sequence are

984
01:00:38,210 --> 01:00:41,360
going to converge or whether
they're not going to converge,

985
01:00:41,360 --> 01:00:44,670
because you don't know what
the rest of them are.

986
01:00:44,670 --> 01:00:48,570
So talking about a rate of
convergence with respect to

987
01:00:48,570 --> 01:00:50,680
the strong law of large numbers

988
01:00:50,680 --> 01:00:53,270
doesn't make any sense.

989
01:00:53,270 --> 01:00:55,880
It's connected directly to
the standard notion of a

990
01:00:55,880 --> 01:01:00,640
convergence of a sequence of
numbers when you look at those

991
01:01:00,640 --> 01:01:04,690
numbers applied to
a sample path.

992
01:01:04,690 --> 01:01:07,900
This is what gives the strong
law of large numbers its

993
01:01:07,900 --> 01:01:13,920
power, the fact that it's
related to this standard idea

994
01:01:13,920 --> 01:01:14,690
of convergence.

995
01:01:14,690 --> 01:01:18,530
The standard idea of convergence
is what the whole

996
01:01:18,530 --> 01:01:22,310
theory of analysis
is built on.

997
01:01:22,310 --> 01:01:26,030
And there are some very powerful
things you can do

998
01:01:26,030 --> 01:01:27,030
with analysis.

999
01:01:27,030 --> 01:01:31,740
And it's because convergence is
defined the way that it is.

1000
01:01:31,740 --> 01:01:35,400
When we talk about the strong
law of large numbers, we are

1001
01:01:35,400 --> 01:01:39,170
locked into that particular
notion of convergence.

1002
01:01:39,170 --> 01:01:41,690
And therefore, it's going
to have a lot of power.

1003
01:01:41,690 --> 01:01:44,050
We will see this as soon
as we start talking

1004
01:01:44,050 --> 01:01:45,750
about renewal theory.

1005
01:01:45,750 --> 01:01:47,890
And in fact, we'll see it in
the proof of the strong law

1006
01:01:47,890 --> 01:01:50,640
that we're going
to go through.

1007
01:01:50,640 --> 01:01:53,470
Most of the heavy lifting with
the strong law of large

1008
01:01:53,470 --> 01:01:57,780
numbers has been done by the
analysis of convergence with

1009
01:01:57,780 --> 01:01:58,740
probability 1.

1010
01:01:58,740 --> 01:02:01,540
The hard thing is this theorem
we've just proven.

1011
01:02:01,540 --> 01:02:02,990
And that's tricky.

1012
01:02:02,990 --> 01:02:05,750
And I apologize for getting a
little confused about it as we

1013
01:02:05,750 --> 01:02:08,720
went through it, and not
explaining all the steps

1014
01:02:08,720 --> 01:02:10,910
completely.

1015
01:02:10,910 --> 01:02:12,950
But as I said, it's hard
to follow proofs

1016
01:02:12,950 --> 01:02:14,950
in real time anyway.

1017
01:02:14,950 --> 01:02:17,080
But all of that is done now.

1018
01:02:17,080 --> 01:02:19,700
How do we go through the strong
law of large numbers

1019
01:02:19,700 --> 01:02:22,990
now if we accept this
convergence

1020
01:02:22,990 --> 01:02:25,370
with probability 1?

1021
01:02:25,370 --> 01:02:29,370
Well, it turns out to
be pretty easy.

1022
01:02:29,370 --> 01:02:32,300
We're going to assume that the
expected value of the fourth

1023
01:02:32,300 --> 01:02:35,890
moment of this underlying
random variable

1024
01:02:35,890 --> 01:02:38,040
is less than infinity.

1025
01:02:38,040 --> 01:02:43,830
So let's look at the expected
value of the sum of n random

1026
01:02:43,830 --> 01:02:47,120
variables taken to
the fourth power.

1027
01:02:47,120 --> 01:02:48,790
OK, so what is that?

1028
01:02:48,790 --> 01:02:57,760
It's the expected value of S
sub n times S sub n times S

1029
01:02:57,760 --> 01:03:00,030
sub n times S sub n.

1030
01:03:00,030 --> 01:03:05,220
Sub n is the sum of
Xi from 1 to n.

1031
01:03:05,220 --> 01:03:07,020
It's also this.

1032
01:03:07,020 --> 01:03:08,840
It's also this.

1033
01:03:08,840 --> 01:03:09,970
It's also this.

1034
01:03:09,970 --> 01:03:14,150
So the expected value of S to
the n fourth is the expected

1035
01:03:14,150 --> 01:03:17,810
value of this entire
product here.

1036
01:03:17,810 --> 01:03:22,800
I should have a big bracket
around all of that.

1037
01:03:22,800 --> 01:03:27,190
If I multiply all of these terms
out, each of these terms

1038
01:03:27,190 --> 01:03:29,900
goes from 1 to n, what I'm
going to get is the

1039
01:03:29,900 --> 01:03:32,710
sum from 1 to n.

1040
01:03:32,710 --> 01:03:35,010
Sum over j from 1 to n.

1041
01:03:35,010 --> 01:03:37,310
Sum over k from 1 to n.

1042
01:03:37,310 --> 01:03:40,810
And a sum over l from 1 to n.

1043
01:03:40,810 --> 01:03:44,710
So I'm going to have the
expected value of X sub i

1044
01:03:44,710 --> 01:03:48,370
times X sub j times X
sub k times X sub l.

1045
01:03:48,370 --> 01:03:50,100
Let's review what this is.

1046
01:03:50,100 --> 01:04:01,340
X sub i is the random variable
for the i-th of these X's.

1047
01:04:01,340 --> 01:04:02,640
I have n X's--

1048
01:04:02,640 --> 01:04:06,310
X1, X2, X3, up to X sub n.

1049
01:04:06,310 --> 01:04:11,000
What I'm trying to find is the
expected value of this sum to

1050
01:04:11,000 --> 01:04:12,460
the fourth power.

1051
01:04:12,460 --> 01:04:15,702
When you look at the sum of
something, if I look at the

1052
01:04:15,702 --> 01:04:34,410
sum of numbers, [INAUDIBLE] of
a sub i, times the sum of b

1053
01:04:34,410 --> 01:04:38,740
sub i, I write it as j.

1054
01:04:38,740 --> 01:04:40,830
If i just do this, what's
it equal to?

1055
01:04:40,830 --> 01:04:45,380
It's equal to the sum over
i and j of a sub

1056
01:04:45,380 --> 01:04:47,890
i times a sub j.

1057
01:04:47,890 --> 01:04:50,650
I'm doing exactly the same thing
here, but I'm taking the

1058
01:04:50,650 --> 01:04:52,270
expected value of it.

1059
01:04:52,270 --> 01:04:55,540
That's a finite sum. the
expected value of the sum is

1060
01:04:55,540 --> 01:05:00,540
equal to the sum of the
expected values.

1061
01:05:00,540 --> 01:05:07,240
So if I look at any particular
value of X--

1062
01:05:07,240 --> 01:05:08,580
of this first X here.

1063
01:05:08,580 --> 01:05:11,390
Suppose I look at i equals 1 .

1064
01:05:11,390 --> 01:05:17,990
I suppose I look at the expected
value of X1 times--

1065
01:05:17,990 --> 01:05:20,460
and I'll make this anything
other than 1.

1066
01:05:20,460 --> 01:05:24,350
I'll make this anything other
than 1, and this anything

1067
01:05:24,350 --> 01:05:24,870
other than 1.

1068
01:05:24,870 --> 01:05:27,370
For example, suppose I'm trying
to find the expected

1069
01:05:27,370 --> 01:05:35,490
value of X1 times X2
times X10 times X3.

1070
01:05:35,490 --> 01:05:38,110
OK, what is that?

1071
01:05:38,110 --> 01:05:42,910
Since X1, X2, X3 are all
independent of each other, the

1072
01:05:42,910 --> 01:05:47,460
expected value of X1 times the
expected value of all these

1073
01:05:47,460 --> 01:05:52,420
other things is the expected
value of X1 conditional on the

1074
01:05:52,420 --> 01:05:54,610
values of these other
quantities.

1075
01:05:54,610 --> 01:05:57,105
And then I average over all
the other quantities.

1076
01:06:00,210 --> 01:06:03,460
Now, if these are independent
random variables, the expected

1077
01:06:03,460 --> 01:06:08,080
value of this given the values
of these other quantities is

1078
01:06:08,080 --> 01:06:11,130
just the expected value of X1.

1079
01:06:11,130 --> 01:06:14,740
I'm dealing with a case where
the expected value of X is

1080
01:06:14,740 --> 01:06:17,800
equal to 0.

1081
01:06:17,800 --> 01:06:20,330
Assuming X bar equals 0.

1082
01:06:20,330 --> 01:06:26,080
So when I pick i equal to 1
and all of these equal to

1083
01:06:26,080 --> 01:06:31,680
something other than 1, this
expected value is equal to 0.

1084
01:06:31,680 --> 01:06:34,760
That's a whole bunch of expected
values because that

1085
01:06:34,760 --> 01:06:39,580
includes j equals 2 to n, k
equals 2 to n, and X sub l

1086
01:06:39,580 --> 01:06:41,090
equals 2 to n.

1087
01:06:41,090 --> 01:06:45,520
Now, I can do this for X sub i
equals 2, X sub i equals 3,

1088
01:06:45,520 --> 01:06:46,770
and so forth.

1089
01:06:48,950 --> 01:06:53,770
If i is different from j, and k,
and l, this expected value

1090
01:06:53,770 --> 01:06:55,020
is equal to 0.

1091
01:06:58,340 --> 01:07:02,160
And the same thing if
X sub j is different

1092
01:07:02,160 --> 01:07:03,430
than all the others.

1093
01:07:03,430 --> 01:07:05,640
The expected value
is equal to 0.

1094
01:07:05,640 --> 01:07:09,150
So how can I get anything
that's nonzero?

1095
01:07:09,150 --> 01:07:15,240
Well, if I look at X sub 1 times
X sub 1 times X sub 1

1096
01:07:15,240 --> 01:07:18,130
times X sub 1, that
gives me expected

1097
01:07:18,130 --> 01:07:19,690
value of X to the fourth.

1098
01:07:19,690 --> 01:07:22,950
That's not 0, presumably.

1099
01:07:22,950 --> 01:07:24,860
And I have n terms like that.

1100
01:07:29,050 --> 01:07:33,350
Well, I'm getting
down to here.

1101
01:07:33,350 --> 01:07:37,540
What we have is two kinds
of nonzero terms.

1102
01:07:37,540 --> 01:07:41,930
One of them is where i is equal
to j is equal to k is

1103
01:07:41,930 --> 01:07:43,060
equal to l.

1104
01:07:43,060 --> 01:07:46,830
And then we have X sub i
to the fourth power.

1105
01:07:46,830 --> 01:07:49,980
And we're assuming that's
some finite quantity.

1106
01:07:49,980 --> 01:07:52,890
That's the basic assumption
we're using here, expected

1107
01:07:52,890 --> 01:07:55,740
value of X fourth is
less than infinity.

1108
01:07:55,740 --> 01:07:58,470
What other kinds of things
can we have?

1109
01:07:58,470 --> 01:08:05,130
Well, if i is equal to j, and
if k is equal to l, then I

1110
01:08:05,130 --> 01:08:13,920
have the expected value of Xi
squared times expected value

1111
01:08:13,920 --> 01:08:16,890
of Xi squared Xk squared.

1112
01:08:16,890 --> 01:08:17,950
What is that?

1113
01:08:17,950 --> 01:08:22,510
Xi squared is independent of
Xk squared because i is

1114
01:08:22,510 --> 01:08:23,710
unequal to k.

1115
01:08:23,710 --> 01:08:26,060
These are independent
random variables.

1116
01:08:26,060 --> 01:08:31,250
So I have the expected value
of Xi squared is what?

1117
01:08:31,250 --> 01:08:35,720
It's just a variance of X.
This quantity here is the

1118
01:08:35,720 --> 01:08:37,819
variance of X also.

1119
01:08:37,819 --> 01:08:43,729
So I have the variance of Xi
squared, which is squared.

1120
01:08:43,729 --> 01:08:50,040
So I have sigma to
the fourth power.

1121
01:08:50,040 --> 01:08:55,720
So those are the only terms that
I have for this second

1122
01:08:55,720 --> 01:08:59,850
kind of nonzero term
where Xi--

1123
01:08:59,850 --> 01:09:02,160
excuse me. not Xi
is equal to Xj.

1124
01:09:02,160 --> 01:09:03,689
That's not what we're
talking about.

1125
01:09:03,689 --> 01:09:06,819
Where i is equal to j.

1126
01:09:06,819 --> 01:09:12,330
Namely, we have a sum where i
runs from 1 to n, where j runs

1127
01:09:12,330 --> 01:09:16,670
from 1 to n, k runs from 1 to
n, and l runs from 1 to n.

1128
01:09:16,670 --> 01:09:21,040
What we're looking at is, for
what values of i, j, k, and l

1129
01:09:21,040 --> 01:09:24,500
is this quantity
not equal to 0?

1130
01:09:24,500 --> 01:09:28,430
We're saying that if i is equal
to j is equal to k is

1131
01:09:28,430 --> 01:09:32,229
equal to l, then for all of
those terms, we have the

1132
01:09:32,229 --> 01:09:34,550
expected value of X fourth.

1133
01:09:34,550 --> 01:09:39,640
For all terms in which i is
equal to j and k is equal to

1134
01:09:39,640 --> 01:09:44,689
l, for all of those terms, we
have the expected value of X

1135
01:09:44,689 --> 01:09:47,609
sub i quantity squared.

1136
01:09:47,609 --> 01:09:50,560
Now, how many of those
terms are there?

1137
01:09:50,560 --> 01:09:55,180
Well, x sub i can be
any one of n terms.

1138
01:09:55,180 --> 01:10:01,220
x sub j can be any one
of how many terms?

1139
01:10:01,220 --> 01:10:02,470
It can't be equal.

1140
01:10:06,634 --> 01:10:11,120
i is equal to j, how many
things can k be?

1141
01:10:11,120 --> 01:10:17,450
It can't be equal to i because
then we would wind up with X

1142
01:10:17,450 --> 01:10:19,550
sub i to the fourth power.

1143
01:10:19,550 --> 01:10:24,650
So we're looking at n minus
1 possible values for k, n

1144
01:10:24,650 --> 01:10:27,430
possible values for i.

1145
01:10:27,430 --> 01:10:30,820
So there are n times n minus
1 of those terms.

1146
01:10:30,820 --> 01:10:32,070
I can also have--

1147
01:10:41,868 --> 01:10:44,600
let me write in this way.

1148
01:10:44,600 --> 01:10:51,580
Times Xk Xl equals k.

1149
01:10:51,580 --> 01:10:52,800
I can have those terms.

1150
01:10:52,800 --> 01:11:00,470
I can also have Xi
Xj unequal to i.

1151
01:11:00,470 --> 01:11:08,720
Xk equal to k and
Xl equal to i.

1152
01:11:08,720 --> 01:11:10,640
I can have terms like this.

1153
01:11:10,640 --> 01:11:13,650
And that gives me a sigma
fourth term also.

1154
01:11:13,650 --> 01:11:18,630
I can also have Xi
Xj unequal to i.

1155
01:11:18,630 --> 01:11:22,370
k can be equal to i and
l can be equal to j.

1156
01:11:22,370 --> 01:11:24,880
So I really have three
kinds of terms.

1157
01:11:24,880 --> 01:11:33,840
I have three times n times n
minus 1 times the expected

1158
01:11:33,840 --> 01:11:43,690
value of X squared, this
quantity squared.

1159
01:11:43,690 --> 01:11:48,190
So that's the total value of
expected value of S sub n to

1160
01:11:48,190 --> 01:11:50,130
the fourth.

1161
01:11:50,130 --> 01:11:55,640
It's the n terms for which i is
equal to j is equal to k is

1162
01:11:55,640 --> 01:12:02,700
equal to l plus the 3n times n
minus 1 terms in which we have

1163
01:12:02,700 --> 01:12:05,790
two pairs of equal terms.

1164
01:12:05,790 --> 01:12:07,470
So we have that quantity here.

1165
01:12:07,470 --> 01:12:13,010
Now, expected value of X fourth
is the second moment of

1166
01:12:13,010 --> 01:12:16,410
the random variable X squared.

1167
01:12:16,410 --> 01:12:23,870
So the expected value of X
squared squared is the mean of

1168
01:12:23,870 --> 01:12:26,610
X squared squared.

1169
01:12:26,610 --> 01:12:30,580
And that's less than or equal to
the variance of X squared,

1170
01:12:30,580 --> 01:12:32,100
which is this quantity.

1171
01:12:32,100 --> 01:12:36,190
The expected value
of Sn fourth is--

1172
01:12:38,980 --> 01:12:43,310
well, actually it's less than
or equal to 3n squared times

1173
01:12:43,310 --> 01:12:45,690
the expected value
of X fourth.

1174
01:12:45,690 --> 01:12:50,020
And blah, blah, blah, until we
get to 3 times the expected

1175
01:12:50,020 --> 01:12:53,300
value of X fourth times the
sum from n equals 1 to

1176
01:12:53,300 --> 01:12:56,140
infinity of 1 over n squared.

1177
01:12:56,140 --> 01:12:59,695
Now, is that quantity finite
or is it infinite?

1178
01:13:06,050 --> 01:13:09,100
Well, let's talk of three
different ways of showing that

1179
01:13:09,100 --> 01:13:13,220
this sum is going
to be finite.

1180
01:13:13,220 --> 01:13:17,710
One of the ways is that this is
an approximation, a crude

1181
01:13:17,710 --> 01:13:20,860
approximation, of the
integral from 1 to

1182
01:13:20,860 --> 01:13:24,395
infinity of 1 over X squared.

1183
01:13:24,395 --> 01:13:26,990
You know that that integral
is finite.

1184
01:13:26,990 --> 01:13:30,560
Another way of doing it is you
already know that if you take

1185
01:13:30,560 --> 01:13:35,650
1 over n times 1 over n plus 1,
you know how to sum that.

1186
01:13:35,650 --> 01:13:37,340
That sum is finite.

1187
01:13:37,340 --> 01:13:41,040
You can bound this by that.

1188
01:13:43,940 --> 01:13:48,570
And the other way of doing it is
simply to know that the sum

1189
01:13:48,570 --> 01:13:50,140
of 1 over n squared is finite.

1190
01:13:53,990 --> 01:13:59,740
So what this says is that the
expected value of S sub n

1191
01:13:59,740 --> 01:14:04,290
fourth over n fourth is
less than infinity.

1192
01:14:04,290 --> 01:14:14,210
That says that the probability
of the set of omega for which

1193
01:14:14,210 --> 01:14:20,450
S to the fourth over n fourth
is equal to 0 is equal to 1.

1194
01:14:20,450 --> 01:14:24,750
in other words, it's saying
that S to the fourth over

1195
01:14:24,750 --> 01:14:30,850
omega over n fourth
converges to 0.

1196
01:14:30,850 --> 01:14:34,200
That's not quite what
we want, is it?

1197
01:14:34,200 --> 01:14:37,770
But the set of sample points
for which this quantity

1198
01:14:37,770 --> 01:14:44,790
converges has probability 1.

1199
01:14:44,790 --> 01:14:47,860
And here is where you see the
real power of the strong law

1200
01:14:47,860 --> 01:14:49,410
of large numbers.

1201
01:14:49,410 --> 01:14:57,170
Because if these numbers
converge to 0 with probability

1202
01:14:57,170 --> 01:15:11,310
1, what happens to the set of
numbers is Sn to the fourth of

1203
01:15:11,310 --> 01:15:15,920
omega divided by n to the
fourth, this limit--

1204
01:15:23,488 --> 01:15:28,940
if this was equal to 0, then
what is the limit as n

1205
01:15:28,940 --> 01:15:35,541
approaches infinity of Sn of
omega over n to the fourth?

1206
01:15:38,880 --> 01:15:43,240
If I take the fourth root
of this, I get this.

1207
01:15:43,240 --> 01:15:47,220
If this quantity is converging
to 0, the fourth root of this

1208
01:15:47,220 --> 01:15:53,830
also has to be converging to 0
on a sample path basis of the

1209
01:15:53,830 --> 01:15:58,300
fact that this converges means
that this converges also.

1210
01:16:00,910 --> 01:16:03,350
Now, you see if you were dealing
with convergence in

1211
01:16:03,350 --> 01:16:07,060
probability or something like
that, you couldn't play this

1212
01:16:07,060 --> 01:16:09,090
funny game.

1213
01:16:09,090 --> 01:16:12,450
And the ability to play this
game is really what makes

1214
01:16:12,450 --> 01:16:16,490
convergence in probability
a powerful concept.

1215
01:16:16,490 --> 01:16:19,350
You can do all sorts of strange
things with it.

1216
01:16:19,350 --> 01:16:23,590
And we'll talk about
that next time.

1217
01:16:23,590 --> 01:16:29,920
But that's why all
of this works.

1218
01:16:29,920 --> 01:16:33,960
So that's what says that the
probability of the set of

1219
01:16:33,960 --> 01:16:37,570
omega for which the limits
of Sn of omega over n

1220
01:16:37,570 --> 01:16:38,820
equals 0 equals 1.

1221
01:16:41,840 --> 01:16:45,590
Now, let's look at the
strange aspect of

1222
01:16:45,590 --> 01:16:47,880
what we've just done.

1223
01:16:47,880 --> 01:16:51,940
And this is where things
get very peculiar.

1224
01:16:51,940 --> 01:16:55,640
Let's look at the Bernoulli
case, which by now we all

1225
01:16:55,640 --> 01:16:57,470
understand.

1226
01:16:57,470 --> 01:17:03,260
So we consider a Bernoulli
process, all

1227
01:17:03,260 --> 01:17:04,860
moments of X exist.

1228
01:17:04,860 --> 01:17:08,110
Moment-generating functions
of X exist.

1229
01:17:08,110 --> 01:17:11,590
X is about as well-behaved as
you can expect because it only

1230
01:17:11,590 --> 01:17:14,030
has the values 1 or 0.

1231
01:17:14,030 --> 01:17:16,630
So it's very nice.

1232
01:17:16,630 --> 01:17:19,690
The expected value of X
is going to be equal

1233
01:17:19,690 --> 01:17:22,070
to p in this case.

1234
01:17:22,070 --> 01:17:28,380
The set of sample paths for
which Sn of omega over n is

1235
01:17:28,380 --> 01:17:31,270
equal to p has probability 1.

1236
01:17:31,270 --> 01:17:37,300
In other words, with probability
1, when you look

1237
01:17:37,300 --> 01:17:40,305
at a sample path and you look
at the whole thing from n

1238
01:17:40,305 --> 01:17:43,730
equals 1 off to infinity, and
you take the limit of that

1239
01:17:43,730 --> 01:17:47,750
sample path as n goes to
infinity, what you get is p.

1240
01:17:47,750 --> 01:17:52,090
And the probability that you
get p is equal to 1.

1241
01:17:52,090 --> 01:17:55,880
Well, now, the thing that's
disturbing is, if you look at

1242
01:17:55,880 --> 01:17:59,930
another Bernoulli process where
the probability of the 1

1243
01:17:59,930 --> 01:18:03,160
is p prime instead of p.

1244
01:18:03,160 --> 01:18:06,630
What happens then?

1245
01:18:06,630 --> 01:18:12,440
With probability 1, you get
convergence of Sn of omega

1246
01:18:12,440 --> 01:18:19,820
over n, but the convergence is
to p prime instead of to p.

1247
01:18:19,820 --> 01:18:24,790
The events in these two spaces
are exactly the same.

1248
01:18:24,790 --> 01:18:28,470
We've changed the probability
measure, but we've kept all

1249
01:18:28,470 --> 01:18:30,740
the events the same.

1250
01:18:30,740 --> 01:18:34,040
And by changing the probability
measure, we have

1251
01:18:34,040 --> 01:18:43,160
changed one set of probability 1
into a set of probability 0.

1252
01:18:43,160 --> 01:18:46,500
And we changed another set of
probability 0 into set of

1253
01:18:46,500 --> 01:18:48,550
probability 1.

1254
01:18:48,550 --> 01:18:52,130
So we have two different
events here.

1255
01:18:52,130 --> 01:18:56,450
On one probability measure, this
event has probability 1.

1256
01:18:56,450 --> 01:18:58,700
On the other one, it
has probability 0.

1257
01:18:58,700 --> 01:19:04,930
They're both very nice, very
well-behaved probabilistic

1258
01:19:04,930 --> 01:19:06,750
situations.

1259
01:19:06,750 --> 01:19:08,560
So that's a little disturbing.

1260
01:19:08,560 --> 01:19:14,140
But then you say, you can pick
p in an uncountably infinite

1261
01:19:14,140 --> 01:19:15,940
number of ways.

1262
01:19:15,940 --> 01:19:18,680
And for each way you
count p, you have

1263
01:19:18,680 --> 01:19:20,160
uncountably many events.

1264
01:19:23,250 --> 01:19:28,790
Excuse me, for each value of
p, you have one event of

1265
01:19:28,790 --> 01:19:31,790
probability 1 for that p.

1266
01:19:31,790 --> 01:19:35,750
So as you go through this
uncountable number of events,

1267
01:19:35,750 --> 01:19:39,480
you go through this uncountable
number of p's, you

1268
01:19:39,480 --> 01:19:43,560
have an uncountable number of
events, each of which has

1269
01:19:43,560 --> 01:19:47,890
probability 1 for its own p.

1270
01:19:47,890 --> 01:19:52,600
And now the set of sequences
that converge is, in fact, a

1271
01:19:52,600 --> 01:19:55,270
rather peculiar sequence
to start with.

1272
01:19:55,270 --> 01:19:57,656
So if you look at all the other
things that are going to

1273
01:19:57,656 --> 01:20:01,010
happen, there are an awful
lot of those events also.

1274
01:20:04,300 --> 01:20:08,780
So what is happening here is
that these events that we're

1275
01:20:08,780 --> 01:20:14,750
talking about are indeed very,
very peculiar events.

1276
01:20:14,750 --> 01:20:16,530
I mean, all the mathematics
works out.

1277
01:20:16,530 --> 01:20:17,850
The mathematics is fine.

1278
01:20:17,850 --> 01:20:21,420
There's no doubt about it.

1279
01:20:21,420 --> 01:20:24,280
In fact, the mathematics
of probability

1280
01:20:24,280 --> 01:20:26,570
theory was worked out.

1281
01:20:26,570 --> 01:20:29,940
People like Kolmogorov went to
great efforts to make sure

1282
01:20:29,940 --> 01:20:31,960
that all of this worked out.

1283
01:20:31,960 --> 01:20:34,330
But then he wound up with
this peculiar kind

1284
01:20:34,330 --> 01:20:36,660
of situation here.

1285
01:20:36,660 --> 01:20:40,170
And that's what happens when you
go to an infinite number

1286
01:20:40,170 --> 01:20:43,080
of random variables.

1287
01:20:43,080 --> 01:20:49,320
And it's ugly, but that's
the way it is.

1288
01:20:49,320 --> 01:20:55,140
So that what I'm arguing here
is that when you go from

1289
01:20:55,140 --> 01:21:00,220
finite m to infinite n, and
you start interchanging

1290
01:21:00,220 --> 01:21:06,010
limits, and you start taking
limits without much care and

1291
01:21:06,010 --> 01:21:09,280
you start doing all the things
that you would like to do,

1292
01:21:09,280 --> 01:21:15,700
thinking that infinite n is sort
of the same as finite n.

1293
01:21:15,700 --> 01:21:18,990
In most places in probability,
you can do that and you can

1294
01:21:18,990 --> 01:21:20,170
away with it.

1295
01:21:20,170 --> 01:21:22,800
As soon as you start dealing
with the strong law of large

1296
01:21:22,800 --> 01:21:26,130
numbers, you suddenly really
have to start being careful

1297
01:21:26,130 --> 01:21:27,810
about this.

1298
01:21:27,810 --> 01:21:31,900
So from now on, we have to be
just a little bit careful

1299
01:21:31,900 --> 01:21:36,120
about interchanging limits,
interchanging summation and

1300
01:21:36,120 --> 01:21:40,190
integration, interchanging all
sorts of things, as soon as we

1301
01:21:40,190 --> 01:21:43,540
have an infinite number
of random variables.

1302
01:21:43,540 --> 01:21:47,600
So that's a care that we have
to worry about from here on.

1303
01:21:47,600 --> 01:21:48,850
OK, thank you.