In the last lectures we discussed sequence
of events , that is Infinite sequence of events
and limits. Limits of these events, limits
lim sup we define, lim sup and lim. Suppose,
sequence is n , this event we defined. Similarly,
lim inf as n tends to infinity, this type
of limit we define and then how to find out
the probability for this limiting events .
We ah discussed different theorems and ah
particularly Borel- Cantelli lemmas ah how
to find out ah this probability, probability
of lim sup A n under two different conditions
. Now ah this application of convergence of
ah Infinite Sequence of events is the ah convergence
of sequence of random variables.
Now, we will be discussing ah the convergence
concepts for random variables and these concepts
are useful in many practical applications
, just like it is the basis of stochastic
calculus . Also ah we will be ah discussing
ah, ah very important applications like Central
Limit Theorem and we log up large numbers.
Ah first of all let us see, what is convergence
of a sequence of real numbers . Consider a
sequence of real numbers x n, n going from
one to infinity . This sequence converges
to a limit x if corresponding every epsilon
greater than 0, we can find a positive integer
N such that x mod of x minus x n can be made
smaller than epsilon for all n greater than
N .
So, what does it say that , ah this difference
between the limit and the ah sequence can
be made as small as possible where alpha epsilon
is an arbitrarily small number positive number
, ah for sufficiently large N. Then we say
that x n converges to x. So, for example,
if I consider this sequence suppose x n is
equal to 1 by n simple example. Now ah, we
know that limit of 1 by n as n tends to infinity
0.
Now, how uh it confirms to this definition
, suppose for any epsilon greater than 0 we
can choose a positive integer N greater than
one by epsilon . So, that in that case 0 minus
ah x n mod of that will be 1 by that is equal
to 1 by n and this will be less than epsilon.
So, because we are considering N. N is greater
than 1 by epsilon therefore, ah for any n
greater than capital N this will be valid
. So, that way ah this 0 will be limit of
this sequence. And now ah we have to extend
this definition of convergence or limit to
ah sequence of random variables.
Consider a sequence of random variable x n,
n going from 1 to infinity defined on S,F,P.
We are defining suppose a sequence of random
variables . Now what does it mean ah; that
means, if we have a sample space suppose S,
this is the sample space and now corresponding
to ah suppose any element s 1 here. So, that
will map to suppose, this is my time axis
n, n is equal to 1 here. So, n is equal to
1 it will be 1 point.
Similarly, n is equal to 2 it may be ah another
point like that ah this s 1 will be mapped
to different points at different instant of
time . So, that way ah I may get a sequence,
may be if I join this may be something like
this . So, that is; that means, corresponding
to every sample point I will get a ah sample
functions like this. So, ; that means, fall
is S belonging to sample space X 1 s, X 2
s up to X n s represent a discrete time sample
function particular for is S . So, that will
be , we will have a sample function like this
.
Therefore, since ah the sample space will
contain other sample points. Therefore, corresponding
to each sample point will get a sample function
like this. So, unlike ah sequence of real
numbers action sequence of random variables
a represents a family of sequences. So, each
sample function is a sequence therefore, it
is a family of sequences and here are the
ah problem in defining the convergence for
the sequence of a real numbers.
So, convergence of a random sequence is to
be defined using different criteria because
it is since it is not a single sequence ,we
have to use different criteria to ah, ah to
the define the convergence of a sequence of
random variables. Therefore there are different
modes of convergence of a sequence of random
variables. The most elementary convergence
concept is convergence everywhere or Point
- wise convergence, it is also called Point
- wise convergence. A sequence of random variables
X n is said to converge everywhere to X if
limit of X n s equal to X of s for all s belonging
to S. So, suppose corresponding to each sample
point we will find a limit .
So, in that case ah this is the point wise
convergent or convergent everywhere ah we
can consider one simple example. Suppose define
a sequence of random variables suppose X only
sample space comprises of two sample points
s 1 and s 2 . Now, ah suppose X n s 1 is equal
to 1 plus 1 by n and X n s 2 is equal to minus
1 plus 1 by n . We observe that as n tends
to it [vocalised-noise] infinity, this point
will become 1 and this point will ah become
minus 1 . So, that way we can ah now define
the a random variable X such that suppose
X of s 1 is equal to 1 and X of s 2 is equal
to minus 1.
So; obviously, limit of X n s 1 will be 1
and limit of X n of s 2 will be minus 1. Therefore,
in this case the sequence X n, this X n sequence
converges point wise or converges everywhere
to random variable X. So, that way hm convergence
ah everywhere is this just an extension of
convergence of a sequence of real numbers
. So, here we are not ah, not considering
the property of the random variable ah, we
are simply considering this as a ah sequence
of function.
Next we will ah, ah consider one concept which
is known as ah convergence almost here or
convergence with probability 1, this is also
known as the strong mode of convergence, this
is strong convergence . . 
Let us consider an event that is it is comprising
of points S such that limit of X n s is equal
to X s. So, we are considering all those sample
points for which limit of X n s is equal to
X s there may be some sample points where
this limit uh ah limit is not equal to Xs
ok, but ah we are considering only those simple
points for which this limit is equal to X
s.
Since decision even define on the on S,F,P
we can find out the probability of this event
. This sequence now X n is said to converge
to X almost here or with probability 1, if
probability of this event that is probability
of the event s such that limit of X n s equal
to X s is equal to 1 . So, probability of
this event is equal to1 ah. So, ah that is
the definition of almost your convergence;
that means, wherever there is a convergence.
So, ah those sample points if we consider
then ah the corresponding event should have
probability 1.
Equivalently for every epsilon greater than
0 there exists N such that a probability of
this event ah is less than epsilon for all
n greater than N, this is the event we are
considering that is S such that X n s minus
X s mod of that is less than epsilon for all
n greater than certain N. Suppose epsilon
corresponding to each epsilon we will get
a ah big N number and ah suppose where this
convergence takes place for all n greater
than equal to N, that probability is equal
to 1 .
We write X n converges to X almost here this
is the symbol we use ah, as I have told this
is a, ah mod of strong conversions ah . Let
us consider one example, suppose sample space
is s 1, s 2, s 3, it comprises of three elements
and the X n be a sequence of random variables
with X n of s 1 is equal to 1, X n of s 2
is equal to minus 1 and suppose this is a
divergent sequence X n of s 3 is equal to
n and now, we will define another random variable
on the same sample space X s 1 is equal to
1, X s 2 is equal to minus 1 and X of s 3
is equal to 1. So, here it is divergent, here
it is 1.
Ah therefore, if I consider the event for
where , ah which this convergence is taking
place I, I we see that X n s 1 is 1 and here
also 1, X n s 2 is minus 1, here also minus
1, but X n s 3 is 1, here it is 1 therefore,
it is not ah converging for s 3. So, that
way this event now where that convergence
is taking place that is that comprises of
s 1 and s 2 ah therefore, probability of this
limiting event now is equal to probability
of s 1 s 2. So, ah when X n will converge
to X almost here if probability of this event
is equal to 1, so that way ah the basic concepts
of almost ah sure convergence is introduced.
Now, how to test almost sure convergence,
X n converges almost sure to X, if probability
of this event s such that limit of X n s equal
to X s, the probability of this event is equal
to 1 or equivalently ah, if this limit is
not equal to X s, if we consider that event
that probability should be equal to 0.
Ah, now let us define a set of divergence
where X n s is not equal to s. So, that event
is we are calling this event as D. Now, ah
that event is equal to ah , the set as such
that mod of X n minus s mod of X n s minus
X s is greater than equal to epsilon and now
infinitely of 1 for all arbitrarily small
epsilon greater than 0. So, what we are ah
defining that the set of divergence is this
set comprising of element such that mod of
X n s minus X s is greater than epsilon infinitely
often for all ah arbitrarily small epsilon
greater than 0 .
So, if suppose there is only divergence in
the case of finite number of n, then we can
always consider the sequence starting with
that number where it is ah diverging suppose
after that the sequence we can consider. So,
that way ah this is important that we define
ah the event s such that mod of X n s minus
X s is greater than equal to epsilon for in
for infinitely many n because if it is finitely
many n then we can always go beyond those
numbers to find out the convergence sequence.
Ah, now for is arbitrarily small epsilon we
can ah find a ah positive integer m, m belonging
to set of natural number, this is the set
of natural number such that 1 by m is less
than equal to epsilon. So, any epsilon, uh
you consider we can get a ah, ah positive
integer like this such that 1 by m is less
than equal to epsilon. So, that we can ah
replace epsilon by 1 by m what is the ah idea
because if it is in terms of m we can enumerate.
Ah so, ah this set of divergence now we can
define as D is equal to the set comprising
of element s such that mod of X n s minus
X s is greater than equal to 1 by m. Instead
of epsilon we are writing greater than equal
to1 by m infinitely often for all m will belonging
to n, ah that is equal to suppose ah now m
is uh we can count m therefore, we can write
in terms of suppose Union of Dm m going from
1 to infinity because all m we have to consider.
So, that way we have define a ah set of divergence
where the random sequence diverges from Xs.
So, that ah we are considering and that is
a union ah Dm because for each time we have
to define that divergence set. So, union Dm
m going from 1 to infinity .
Ah, now ah we ah get a condition for s convergence
[vocalised-noise] convergence in terms of
D . Suppose this set of divergence can be
rewritten as ah which we have already done
that it is a Union of Dm m going from 1 to
infinity. Now this infinitely often that is
the, that is the lim sup of s such that mod
of X [vocalised-noise], X n s minus X s is
greater than equal to1 by m as n tends to
infinity, that is the ah event of this ah,
that X n minus X n s minus X s absolute value
of that is greater than 1 by m infinity often
that we are writing in terms of lim, lim sup.
Therefore, ah now this is equal to as ah from
definition of lim sup intersection n going
from 1 to infinity ah [vocalised-noise] of
Union there going from n to infinity of the
event s such that mod of X s minus X s is
greater than equal to 1 by m . So, this is
the basic event where it is ah greater and
this deviation is greater than 1 by m then
we are considering the lim sup of that event
sequence.
Ah; obviously, now X n will converge to X
almost sure if and only if probability of
Union of Dm is equal to 0. So, we ah we have
the condition now if and only if condition
that ah X n will converge to X almost sure
if and only if probability of Union of Dm
m going from 1 to infinity is equal to 0,
this is a very strong condition .
Ah now we will ah because here we have the
Union, Union of Dm m going from 1 to infinity,
this condition we want to simplify. We will
state the theorem ah the sequence X n converges
almost sure to X if and only if probability
of Dm is equal to 0 for each positive integer
m. So, this is a simpler condition because
we have to consider ah P of Dm instead of
the probability of Union Dm .
Now, ah as the proof suppose X n converges
to X , almost here then probability of Union
of Dm m going from 1 to infinity must be equal
to 0 but we also know that this Dm ,any Dm
any member is a subset of the Union, Union
D i, i going from 1 to infinity.
Therefore, we can write that probability of
Dm because it is a subset it is a less than
equal to probability of this Union but ah,
we know that probability of this Union is
equal to 0 therefore, probability Dm must
be equal to 0 . So, that way if X n converges
almost sure to X, then probability of Dm must
be equal to 0 .
Now, next if probability of Dm is equal to
for all m greater than equal to 1 then probability
of a Union of Dm , I know that that is less
than equal to ah sum of D corresponding probability,
this inequality we proved earlier ah therefore,
and I know that each of this probability is
equal to 0 therefore, probability of this
Union will be equal to 0 ..
So, that way we have proved that X n converges
to X if and only if probability of D m is
equal to 0 for each positive integer m ah
so, this is a ah condition that is, uh if
and only if condition ah probability of dm
is equal to 0 gives a necessary and sufficient
condition for almost your convergence.
So, we can ah find out this probability to
prove ah almost your convergence but ah this
is still a , ah difficult task because we
have to consider Dm then Dm is ah, Dm for
each m and we know that Dm is defined through
that lim sup operation . Ah now we will state
a theorem ah that is a which is a consequence
of ah first Borel- Cantelli lemma ah that
gives a sufficient s for almost sure convergence
that we will discuss now.
Ah, theorem if for each m greater than equal
to 1, now we are considering this sum probability
of mod of X n minus x greater than equal to
m . So, ah this is for each n ,we will consider
and then sum up. So, therefore, infinite sum
of probability of mod of X n greater than
equal to 1 by m . So, if this infinite sum,
sum of the infinite series is less than infinity,
then X n converges almost sure to X. So, we
have to ah consider the probability of deviation
for each m and then sum up, if this sum is
less than infinity then X n converges almost
sure to X. Suppose ah, ah for each m greater
than equal to 1this sum is convergent.
So, what does it mean? Now, ah we can ah apply
the Borel - Cantelli lemma if the sequence
of event .suppose I can consider this as a
sequence of event , if the probability of
this sequence if the sum of the probability
of this sequence is less than infinity , then
a probability of lim sup of that event sequence
ah is equal to 0, that is the Borel Cantelli
lemma 1.
So, what we will get here this implies that
probability that lim sup of ah, ah the event
s such that mod of X n s minus X, that is
greater than 1 by m as n tends to infinity
is equal to 0, that is the conclusion. We
ah draw with the help of Borel-Cantelli lemma
1 .
Therefore, , but this is by definition this
is the Dm therefore, probability of Dm is
equal to 0 the probability of Dm is equal
to 0, then X n converges almost sure to X.
So, that way we can apply the ah Borel- Cantelli
lemma ah 1 and that is first part of Borel
-Cantelli lemma to prove the ah, prove the
convergence of a sequence and thus we can
apply this condition ah that is ah probability
then mod of X n minus X greater than equal
to 1 by m , that is the sum from n going from
infinity , ah if that sum is convergent then
X n ah converges to X almost sure but this
is your sufficient ah condition unlike probability
of Dm is equal to 0 that is a necessary and
sufficient condition but this is an this is
a sufficient condition only.
Ah we shall consider one example suppose the
random sequence X n is given by X n is equal
to 0 with probability 1 minus 2 to the power
n and equal to n with probability ah 1 by
2 to the power n. So, we see that , ah this,
this probability will become 1 as n tends
to infinity and correspondingly this probability
will become 0 . So, therefore, X n approaches
0 .
Now, whether this is the limit we get in the
almost your ah sense let us see . So, ah,
ah we see that probability of X n mod of X
n minus 0 greater than equal to 1 by m that
is equal to probability of X n is equal to
n because for n only ah it is this quantity
is becoming ah different from 0 here it is
exactly 0. Therefore, only n we have to consider
probability X n is equal to n that is equal
to 1 by 2 to the power n .
So, we have to consider this sum now, sum
of this probabilities. So, if we consider
sum up from n is equal to 1 to infinity that
is equal to sum of 1 by 2 to the power m as
n going from 1 to infinity and we know that
this sum is less than infinity this is a ah
geometric series we can find out this sum.
So, this sum is less than infinity. Therefore,
ah, ah these ah sequence X n converges almost
sure to X. So, that way ah, we can ah prove
whether a sequence convergence ah to uh a
to a random variable X in the almost sure
sense.
Ah, let us summarize the lecture, ah recall
that this X n this is sequence of random variable,
it represents a family of sequence. So, unlike
ah in the case of sequence of real numbers,
the sequence of real random variables represents
a family of sequence.
Convergence of a random sequence is to be
defined using different criteria the sequence
is said to converge everywhere to X or point
wise point, point wise converge to X if limit
of X n s is equal to X s for all s belonging
to S, that is the ah convergence everywhere
.
Ah now this sequence X n n going from 1 to
infinity is said to converge to X almost sure
or with probability1 if probability of ah
this event that is S such that limit of X
n s equal to X s that is equal to 1. So, this
is the, ah definition of almost sure convergence
which we have introduced here and this is
a strong mode of convergence. And then ah,
ah for testing almost sure convergence we
consider this event Dm, what is the Dm lim
sup of ah the event S such that xjs minus
s X s mode of that should be greater than
equal to 1 by m.
So, lim sup of this event and this is given
by this definition of lim sup only we are
considering here .Ah now, ah we have a necessary
and sufficient condition X n converges almost
0 to X if and only if probability of Union
of Dm m going from 1 to infinity is equal
to 0. The test is simplified ah that is X
n converges ah almost sure to X if and only
if probability of D m is equal to 0 for each
positive integer m .
Now, using the first Borel - Cantelli lemma,
we got a sufficient contest for almost sure
convergence ah if for each m greater than
1, this sum n is equal to1 to infinity probability
of mod of X n minus x greater than 1 by m.
So, the, if the sum is less than infinity
,then ah X n sequence X converges almost 0
to X.
Thank you .
