We are going to now start understanding what
we mean by convergence of a sequence of random
variable. So, we already known like when you
have a given sequence what it converges to
and the limits are very important because
many of the times like integral differential
functions are all defined in terms of the
limits. And also when I have a process, let
us say I have modeled it as some stochastic
process, which are indexed by time. I want
to understand how this process evolves over
time.
So, I would be interested in knowing as I
let my t go to infinity. How my process evolves
and behave? Suppose, you are you are investing
your money in a gamble, every day you are
going to win or lose and based on that every
day you have some money left with you. And
you want to understand eventually if I continue
to play the same game, eventually as the number
of plays becomes large, will I end up with
a positive amount with me or it will all go
to zero or it becomes negative.
So, you want to understand as time evolves
in my stochastic processes as the index becomes
large, how my process looks like, is there
any limiting behavior in that. But then the
question is; fine, we understand what we mean
by convergence of a sequence of random variable.
Sorry, convergence of a sequence of numbers,
then what does we mean by convergence in random
variables? So, we need to make that notion
of convergence of a sequence of random variables
precise and that is what we will focus on
maybe in the next two to three classes.
So, let us say I have a sequence of random
numbers, random variables, all random variables
defined on the same probability space. So,
another thing is I am going to only focus
on are random processes here or a sequence
of random variables, which are indexed by
discrete, which are indexed by discrete numbers.
So, here n is one, two, three, four like that.
Now, the first notion of, sorry, convergence
in almost sure sense. We are going to say
the sequence of random numbers Xn they converge
almost surely, if probability of limit extends
to infinity of this X is going to take the
value X. So, what I mean by this limit here,
I have already said that limit of Xn is equals
to X.
So, what I mean by this here? Suppose, you
fix omega belongs to omega and then you are
going to look at Xn of omega: Is this is a
deterministic sequence Xn of omega? Yes, if
I fix an omega this will denote, because Xn
is a function of from capital omega to R.
So, if you fix an omega, this is some real
number. And I have a sequence of such real
numbers. And I know what, what I mean the
convergence of this sequence of real numbers,
that is a standard convergence and whatever
is the limit.
Now, what we are saying is if the meaning
of this is if a limit of Xn of omega so, so
this is the short form for this. I am going
to look at omega and see whether the sequence
of Xn omega converge to my limited random
variable and then look, if wherever that convergence
happens, I am going to look at all this omegas
and then look at whether that probability
is equals to one. If this happens, then I
am going to call my Xn converges to X almost
surely.
So, let us look an example. So, let us say
I am going to define a sequence of random
variables like this. So, before that, I want
to introduce this notion of unit interval
probability space. So, earlier we have already
defined the notion of what is event space?
What is sigma algebra? And what is what we
mean by probability space and all? So, now
I want to like define a special probability
space, which is as following that omega is
that omega is. So, omega is basically interval
and then this outcome, omega is drawn from
omega, with preference, with no preference
towards any subset.
In a way like I am saying that I am going
to draw omega from this sample space in a
uniformly, without giving any preference to
any of this subset. Then I am going to look.
I am going to now start constructing my event
space on this let us say F, to include all
intervals. What I mean by this? I am going
to say that let I am going to take. So these
are all possible intervals right take a and
b and which is a is between 0 1 and also b
is between 0 1 this is going to define one
interval and let us all possible intervals
be contained in this event space okay.
Now, that I have been drawing omega from this
capital omega without any preference to any
of this subset, then one natural way I am
going to assign probabilities to this interval
is b minus a. I assume that b is going to
be greater than or equals to a. Now, I have
defined omega, I have defined my F partially
here because I only said it includes closed
this intervals here, but if it has to be a
sigma algebra, then I know that all its, it
has to satisfy the properties of sigma algebra,
which says that compliments should be there
and their unions, finite, finitely many and
countable unions should all be there.
So, if intervals are there then the open sets
are also there in this sigma algebra F right.
I will have open intervals in this, I will
have closed intervals in this, by taking the
unions I will have sets which are open at
one end and closed at other, other end and
I will have all such kind of combinations,
if I have to look at a sigma F.
Now, when I have a such a sigma then to make
this completely probability space, I also
need to say how I am going to define probability
for each of the elements in my F. For intervals
it is easy. I have defined like that. But
if I am going to look at all possible elements
that are in F, how I am going to define it.
Yeah, say if you are to take b equals a, then
you have only one element and for which are
saying 0. I am saying you take any interval
here for which you have defined like this
and you can go like this and if you want this
F to be sigma algebra, it will have all possible
subsets of my 0 1 interval. So, the worst
case, what you can take one, we can take this
F to be a power set of this that means it
includes all subsets of the interval 0 1.
But here is some technicality here when you
do this it’s so happens that we will end
up with certain sets for which we will not
be able to consistently assign probabilities.
I mean that comes from some complicated analysis
or some better understanding of the real sets
but we will not go in to that, but what we
are going to take is when this F to include,
what we are going to take this is F to be
the smallest sigma algebra containing all
sub intervals of omega.
So, we take F to be the smallest sigma algebra.
So, you can come up with many-many sigma algebra
which will contain all these intervals. But
we were going to take that sigma algebra to
make this, to define this, which is the one
which contains, thus, which contains all sub
intervals of omega. Is this point clear? So,
we are going to take all, so the way we are
defining F is let it include all the intervals
to make it sigma algebra, we also has to allow
it to possibly include all open sets, their
unions their intersection, so many combinations
are there.
So, you may end up with so many sigma algebras,
which so many F which may it on its own satisfied
the properties of sigma algebra, but among
all them, we will take the one to include
the smallest one, which includes all the sub
intervals. So, this is just to make this bit
more formal and then such a, when we have
such an F which contains all the smallest,
all these sub intervals, we know how to assign
probability to them and using that we can
up try to define the probability for each
of the element in that F.
And such probability space we are going to
call it as unit interval probability. So,
for all practical purpose what we mean by
unit interval probability space is my sample
space is unit interval and my sigma algebra
is such that it contains all possible intervals
and on each of the intervals there, I am going
to assign probability like this if I a interval.
So, we only need to take, so our understanding
of unit probability space will be just this.
So, but it has something more to it in terms
of how this F is defined in terms of the smallest
sigma algebra containing all the sub intervals.
Fine. This one, just a detail and this is
what our understanding of unit interval probability
space.
Now, to understand our notion of convergence,
we will be looking at examples are random
variables defined on this unit interval probability
space because this is going to be easier for
us to understand. So, I am going to take let
omega F P be what I call as unit interval,
probability space. Okay. Now, I am going to
define my random variables like this, fix
an n and that n is going to be such that for
any omega that is coming from capital omega,
it is going to be defined like this. Now,
let us try to map this.
So, let us say this is X1 and this is my omega.
So, how does X1 look like? It is going to
be linear curve? It is going to be linear
and going hitting at 1. And how does X2 look
like? It is like quadratic. And how does X3
look like? So, its curvature will open up
this side or that side?
right side.
It is going to be 
like this. Now, let us try to apply this definition
here and see where it will converge. So, let
us before this, as we saw that as we move
from one two three, the curvature is opening
towards the right, and as n goes to infinity,
how does this curve look like?
So, it is going to almost 0 till this point
and that omega equal to 1 it is going to be
1. So, in this case in a way as n is tending
to infinity we see that this sequence of graphs
here are converging to a place where it is
all 0 and then it suddenly shoots up to 1
at omega equals to 1. So, let us take that
to be our limiting X.
So, then let us try to analyze whether this
property holds and in this case can be call
Xn converges to that X. So, so claim is, we
want to check whether Xn converges to X in
almost surely, where X is at omega equals
to 1 this is my X. So, now can you verify
and see whether this guy satisfies this property?
Whether this is true or wrong? Yeah?
So, let us take our omega which is not at,
so let us take omega which is not 0, not 1.
If you take any omega not 1 that means it
is strictly less than 1, what is going to
happen Xn of omega will converge to what?
It is going to converge to 0? And X is also
0 in that range and if you take omega equals
to 1, what is this going, guys are going to
convert Xn of omega, they are going to be
1 and what is this guy is, this is going to
be 1.
And then it looks like curve, so all omega
which are between 0, 1 are going to satisfy
this property. So, they are included in the
set and what is the probability of that set?
What is the probability that set, 1 because
every point in omega in omega has satisfied
this and so, the probability of big omega
is one that we already know.
So, by this thus, by this definition, we already
know that this example converges to X which
is like this, almost surely. Just let me check
this at the point omega equals to 1 at omega
equals one this guys, these guys are all 1
1 1 and so...
So, fine if I have defined my X equals to
like this, which is 1 only at omega equals
to 1. So, let us say I am going to define
my X to be in a slightly different fashion.
I am going to take my X to be 0 all the way
even at omega equals to 1. It is not jumping
at all here. Is it for that in this case?
Let me call this as X prime here. Is it true
that my Xn converges to X prime almost surely?
No
Why?
Yeah, but I do not care about one point, what
I care about this probability. So, as you
said, other than omega equals to 1 everywhere
this holds, only omega equals to 1 not included.
So what is the probability of this set? We
all omegas are included except omega equals
to 1. It is going to be still 1? Because probability
of that singleton 1 is going to be 0.
Is it true that in that case my Xn converges
to X prime also here because that singleton
values did not have any mass in our example
here. So, both like, both are valid, like
I can say that both converge to this random
variables. So that is fine. Another thing
we will like away notice is, when we have
a deterministic sequence of random variables
whether my limit was always unique.? Was it?
Like, is it possible to, let us say I have
a deterministic sequence an, can it have two
limits? So, then limit is always unique.
But when you are talking about convergence
of this random variable that is not the case.
So, here X is the, when I said Xn convergence
to X, this is my limiting random variable
and these are my sequence of random variables.
So, it is fine that is not the case that I
have only one unique random variable. So,
but as you will see that this is only at the
points which carry 0 mass. So in that way,
this random variable and this random variable
on this probability space they are identical,
because they only differ at points which has
0 mass.
