.
In this model, ah we are going to discuss
a limiting distributions. So, we cover three
ah important topics; ah um first topic is
called modes of convergences and the second
topic we are going to discussed we are going
to discuss law of large numbers and the third
topic which we are going to discuss, that
is a very important topic, that is central
limit theorem .
In this model, we are going to discuss ah
these three ah topics has a three different
lectures. In the first lecture, we are going
to discuss modes of convergences; second lecture,
we are going to discuss law of large numbers;
third lecture, we are going to discuss central
limit theorem . The first lecture is on modes
of convergences .
Till now what we discussed the probability,
space probability of a event, then we discussed
a conditional probability of events, then
we have introduce a random variable, then
we discussed the CDF of the random variable.
Then, based on the discrete type or continuous
type random variable we discussed the probability
mass function, probability density function
.
After we introduced one random variable in
a probability space we have discussed the
distribution in the form of a CDF ah density
function or mass function . Later, we said
one random variable is not enough to solve
some particular problems, you may need more
than one random variables to be defined in
the same probability space to solve the given
problem .
Then we introduce two random variables then
we discuss the CDF, we discussed the joint
probability mass function, we discussed joint
probability density function, then later we
is discuss the conditional probability density
function, conditional probability mass function
for the first random variable, we discuss
the mean variance and so on, that I missed
earlier second order moment in a third other
moment and so on and here we discuss the conditional
distribution, conditional probability conditional
expectation and so on .
Not only two random variables, the later we
introduce n random variables in the same probability
space, then we discuss the joint distribution
of n dimensional random variable, after that
we introduce the transformation from 1 dimensional
to another n dimensional random variable or
r dimensional random variable and so on.
Now, we are going to discuss not a 1 random
variable, not 2 random variables not ah n
finite random variables we are going to discuss
sequence of random variables . This is also
possible when you solve a given problem you
may need to know the sequence of random variable
and what is their distribution as n tends
to infinity? .
The first question is comes whether if you
have a sequence of random variables whether
if you have a sequence of random variable
whether this sequence of random variable converge
or not converges to one random variable or
not if it converge what is the distribution
of that sequence of random variable that is
going to be the topic of ah modes of convergence
for the random variables .
For these first we can understand how the
sequence of a real number real numbers converges
suppose you have real numbers a 1 a 2 and
so on a n and so on whether this sequence
of a real numbers converges or not if it is
converges; that means, a n converges to say
a what is the value of a, that we have discussed
in the any real analysis courses of a sequence
and series of real numbers and so on .
The same concept the we are planning to introduce
for the sequence of random variables , but
the only difference is the random variable
takes a real numbers with some distribution,
that means; ah the X 1 random variable can
take the real values x 1 with some distribution.
Similarly, the random variable X 2 may takes
a random ah may takes a values a real values
x 2 with some distribution and so on. Similarly,
the random variables X n can take the real
number x n with some distribution .
If you know the distribution of a each random
variable X 1, X 2 and so on you have a sequence
of random variable all are defined in the
same probability space, that is very important
all are defined in the same probability space
and if you know the distribution of ah this
sequence of random variables whether this
sequence of random variable converges to one
random variable or not.
If it converges what is the distribution of
that that means, if I write X n converges
to X this is notation am I write X n converges
to X; that means, I have a sequence of random
variable X 1, X 2, X 3 and so on whether this
sequence of random variable converges to the
one random variable that is call it as X.
If I know the distribution of X size what
could be the distribution of x? that is the
question.
In sequence of real numbers converges to one
real number a that may be easy comparing to
the sequence of random variable converges
to the one random variable. Since, each random
variable attached with some distribution for
some random variables moments of a first order
may exist further moment may not exist and
so on . Therefore, you cannot make a only
one way you can conclude ah this sequence
of random variable converges to one random
variable x. There may be more than one ways
you can conclude that this sequence of random
variable converges to one random variable
that we call it as a modes of convergence
.
So, we are going to discuss there are 4 modes
of convergence the first mode of convergence
we write it as X n converges to X in distribution
by writing small letter d above the arrow
I am going to give the definition one by one
in detail with examples. Comparing to the
sequence of real numbers how it converges
to one random variable. Similarly, we are
going to discuss how the sequence of random
variable converges to one random variable
in different ways the different ways we say
it as a different modes.
The first mode of convergence that is called
the convergence in distribution with the letter
small d above their the second one that is
convergence in probability the third one convergence
in r-th moment, the forth one convergence
in almost surely a dot s, above the arrow
if you write d that means, convergence in
distribution, above the arrow if I write small
p that means, convergence in probability,
above the arrow I put the slash with r; that
means, ah it is a convergence in the r-th
moment the forth one convergence in almost
surely .
So, these are all the four ways the sequence
of random variable different in a same probability
space converges to one random variable which
we denoted as a capital X that is also defined
in the same probability space in four different
modes of convergence .
Let us start with the first mode of convergence
that is convergence in distribution 
in distribution let X 1 X 2 and so on X n
and so on be a 
sequence of random variables with CDF F 1,
F 2, and so on F n respectively that means,
the random variable X 1 has a CDF F 1 the
random variable X 2 has a CDF F 2 and so on.
All these random variable defined in the same
probability space.
We say that we say that X n the sequence n
take the value 1, 2 and so on converges in
distribution 
to the random variable denoted by capital
X can be written as X n converges to X in
distribution if limit n tends to infinity
F n of x that is same as F of x for all x
where, where F is the CDF of the random variable
X 
as long as the limit n tends to infinity.
F n of x is same as F of x where F of x is
CDF of the random variable X that is valid
for all X this condition valid for all X then
we can conclude the sequence of random variables
convergence in distribution to the random
variable X note that a F 1, F 2, F n and so
on that is the CDF of the sequence of random
variables respectively and that converges
to a function that is the CDF of the random
variable X, then we can conclude ah this convergence
in distribution to the random variable X .
Let us give a one simple example through that
we can understand the definition clearly.
Example; let omega, F, capital P be a probability
space . 
Let X n; n is equal to 1, 2, and so on be
a sequence of random variables defined on
the probability space omega, F, P that is
defined as X n is defined from omega to R
such that such that X n of W that is equal
to 1 by n for n is equal to 1, 2 and so on
.
So, we are defining the sequence of random
variable from omega to R such that X n of
the W takes a value 1 by n for n is equal
to 1, 2, and so on . Now, we will find out
what is ah CDF of a this sequence of random
variable interview. So, if you find out the
CDF of the n-th random variable as a function
of x this is going to take the value 0, when
x is going to be lesser than 1 by n from 1
by n onwards when x is going to be 1 by n
onwards it is going to take the value 1 . So,
this is going to be the CDF of the sequence
of random variable x size. So, here n takes
a value 1, 2 and so on .
Now, let us go for finding the limit n tends
to infinity of F n of x what could be the
value as limit n tends to infinity for the
F n of x this is going to be 0 when x is lesser
than 0 because as n tends to infinity of 1
by n that becomes 0 and 1 from 0 onwards.
This is a limit n tends to infinity of F n
of x .
Suppose, I denote this as the F of x suppose
suppose I make F of x that takes a value 0,
when x is lesser than 0, 1 from 0 onwards
verify whether this is going to be the CDF
of some random variable it start from 0 land
up 1 and so on. It is satisfies all the properties
of a CDF therefore, this is the F of x is
the CDF of some random variables you denote
it as a capital X say F of x is the CDF of
the some random variable X by seeing limit
n tends to infinity of F n of x that is same
as F of x .
Since limit n tends to infinity F n of x that
is same as F of x, where F of x is the CDF
of some random variable x and the left hand
side F n of x is the CDF of the sequence of
random variable or F n of x is the CDF of
the random variable X n as the limit n tends
to infinity that is same as the CDF of the
random variable x. Therefore, we can conclude
this is the condition is satisfied by the
convergence in distribution.
Therefore, you can conclude the X n converges
to X in distribution . There is a possibility
the sequence of random variable CDF's may
converge to some function that may not be
the CDF. So, as long as ah this sequence of
ah CDF converges to some function that is
also the CDF of some random variable then
you can conclude the X n converges to X in
distribution . So, like that we have some
more problem that we will discuss little later
; that means, when we discuss other mode of
convergence we can verify whether this satisfies
a convergence in distribution also.
So, now will move into next mode of convergence
that is a convergence in probability. Let
X 1, X 2, X n and so on be a sequence of random
variables defined on the probability space
omega, F, P . We say that this sequence of
random variable converges in probability 
to the random variable capital X and write
it as X n converges to X in probability.
If for any epsilon which is greater than 0,
limit n tends to infinity probability of absolute
of X n minus X greater than epsilon that is
equal to 0 . If this condition is satisfied
for any epsilon greater than 0 finding out
the probability in absolute is X n minus X
greater than epsilon limit n tends to infinity
is going to be 0, then we conclude the this
sequence of random variable converges in probability
to the random variable X .
Note that to verify this sequence of random
variable converges in probability you should
know the random variable X, then finding of
the probability after is verified then you
can conclude this sequence of random variable
convergences in probability that means, you
should know about the distribution of the
random variable X or at least you should know
how to compute the probability of absolute
of X n minus X greater than epsilon for any
epsilon greater than 0 that means, beforehand
you should have a the distribution of the
random variable X along with the distribution
of the sequence of random variable X n's then
only you can conclude whether this sequence
of random variable convergence in probability
.
So, for this mode of convergence will go for
the example, through that we will understand
. The example is a example let X n be a sequence
of random variables defined on omega, F, capital
P such that the probability of X n is equal
to 0 that is 1 minus 1 divided by n and the
probability of X n takes a value n it is 1
divided by n. So, this is true for all n,
n is equal to 1, 2, and so on .
We have a sequence of random variable whose
distribution is defined X n takes a value
0 with the probability 1 minus 1 by n or X
n takes a value n with the probability 1 by
n; that means, this sequence of random variables
are of the discrete type which has the two
points, 0 and n are the mass points either
the mass is at 0 or n for the n-th random
variable and you have sequence of random variable
n is equal to 1, 2 and so on .
In this example we can go for taking let epsilon
greater than 0 you can go for finding probability
of absolute of X n which is greater than epsilon
finding out the probability of absolute of
X n greater than epsilon that is same as this
is going to be 1 by n if a epsilon is going
to be lesser than n. If epsilon is going to
be greater than or equal to n the probability
of absolute of X n greater than epsilon is
0 this is for fixed epsilon is greater than
0 .
Now, you can go for taking a limit n tends
to infinity of probability of absolute of
X n greater than epsilon . 
As a limit n tends to infinity this quantity
is going to be the right inside is going to
be 0. Since the limit n tends to infinity
probability of absolute of X n greater than
epsilon is 0 therefore, you can treat the
absolute of X n minus 0 that is equivalent
of concluding X n converges to 0 in probability.
0 you can treat it as the a random variables
X takes a value 0 with the probability 1 . You
can make a X n tends to X in probability,
where X is a degenerated variable or constraint
which takes a value 0 with the probability
1 .
So, sometimes are the sequence of random variable
converges to constraint also. So, this is
example in which we have given sequence of
random variable converges to constant 0 in
probability. Since it is a converges to 0
therefore, we are directly going for probability
of absolute of X n greater than epsilon sometimes
if you have a random variable X and whose
distribution is known then you can go for
finding out the probability of absolute of
X n minus X greater than epsilon then whether
the limit n tends to infinity this quantity
is going to be 0 or not accordingly you can
conclude sequence of random variable converges
to random variable X in probability or not.
So, here it is a very easiest example in which
we are land up the sequence of random variable
converges to 0 in probability .
