Now, we will discuss the third mode of convergence
that is convergence 
in rth moment. Let, r greater than 0 x1, x2
and so on xn and so on be a sequence of random
variables defined on omega F capital P such
that the rth moment for the sequence of random
variable is exist; the absolute sense expectation
is going to be finite for n is equal to 1,
2 and so on.
We say that, we say that this sequence of
random variable converges to the random variable
X in rth moment, if the expectation of the
random variable X the rth moment that is finite
and X also defined in the same probability
space of course.
And limit n tends to infinity, limit n tends
to infinity expectation of in absolute sense
X n minus X to the power r that expected quantity
is going to be 0. We are keeping r greater
than 0 and we have a sequence of random variable
defined in the probability space such that
rth moment exist for those sequence of random
variable.
We say that the sequence of random variable
converges to X in rth moment provided the
rth moment exist for the random variables
X which is defined in the same probability
space and the limit n tends to infinity expectation
that difference of this random variable, the
rth moment is going to 0.
If these conditions are satisfied, then we
can conclude the sequence of random variable
converges to the random variable in the rth
moment.
We can go for the example to explain this
mode of convergence also. That is let X1,
X2, Xn and so on be a sequence of random variables
with mean mu ok, identical mu and variance
of each random variable that is sigma square
or exist both exist and is denoted by mu for
the mean and variance is sigma square. I am
going to define the new random variable that
is sum of n random variables divided by n;
sum of random variable we usually use the
notation S n.
But now, I am going to divide that sum divided
by n. So, I am going to use a notation called
X bar. X bar is the random variable, it is
a function of n. I am not writing n in the
left side, X bar is the sum of random variables
divided by n fine or we can another notation
we can use as a function of S. S n divided
as a function of n. S n divided by n. Either
I can use X bar or S n divided by n, both
are one and the same that is sum of random
variable. So, this is the .
So, I am going to define this random variable
for different n. Will go for finding first
expectation of expectation of absolute of
S n by n minus mu the whole square. In absolute
sense, we will find what is the value of S
n by n minus mu square in absolute sense that
is same as expectation of S n is the square
S n minus n mu divided by S n minus n mu whole
square divided by n square.
That is same as , I can take 1 by n square
outside, if you see S n minus n mu whole square
expectation. If you find out the expectation
of S n expectation of S n that is the expectation
of X1 plus X2 plus Xn all the expectation
of Xi's are same that is mu. Therefore, it
is n mu.
So, since expectation of S n is n mu . So,
the expectation of S n minus n mu whole square
is nothing but variance of S n; that is same
as 1 divided by n square. I will make a additional
condition X is are sequence of independent
random variable with the mean is mu and the
variance is sigma square.
Therefore, the variance of S n is nothing,
but variance of X1 plus X2 plus so on plus
Xn and variance of X is or sigma square. Therefore,
it is going to be n sigma square. Therefore,
this is going to be sigma square by n. We
got the expectation of absolute of S n by
n minus mu whole square that is going to be
sigma square by n. Now, let me apply limit
n tends to infinity of expectation of absolute
of S n by n minus mu the whole square. Since,
n is in the denominator this becomes 0 .
So, this is the definition we wanted to convergence
in rth moments; that means, I can conclude,
S n by n converges to mu in second order moment.
The left right hand side it is not a random
variable, it takes a value mu; that means
it is a random variable takes a value mu with
the probability 1.
So, Xn converges to rth moment with number
r. In notation, we write S n by n converges
to mu in the second order moment. In the right
hand side mu means it is a random variable
takes a value mu with the probability 1. I
have already written X n by n as the X bar
. So, X bar converges to mu in second order
moment.
In statistics we use the X bar that is sum
of random variable divided by n that is from
the n random variable we call it as a Sample
mean. The sequence of random variable having
the mean mu, variance sigma square. So, the
mu is called Population mean . So, the conclusion
is if you have a population with the mean
mu and if you get a sample of size n, then
the sample mean will converge to the population
mean as n tends to infinity.
That means, if you for a large sample or a
large sample the sample mean will converge
to the population mean and this converges
takes place in the convergence in rth moment,
where r is equal to 2 here. Because we are
finding the second order moment convergence
is used. Therefore, sample mean converges
to the population mean in second order moment.
We move to the fourth mode of convergence.
Convergence in Almost Surely. 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 the sequence converges, the sequence
converges in almost surely to the random variable
X, if the probability of limit n tends to
infinity. X n is equal to X that is 1.
So, here the random variable X is defined
in the same probability space. The probability
of limit n tends to infinity X n is equal
to X that is going to be 1, then one can conclude
the sequence of random variable convergence
to the random variable X in almost surely.
So, we write it X n converges to X above the
arrow, you write a dot S ; that means, this
convergence in almost surely.
Suppose, we make a suppose we make a event
A is nothing but collection of w's belonging
to omega such that X n of w tends to X of
w. In that case when I say X n converges to
X almost surely, if and only if the probability
of the event A that is going to be 1. If I
make a event A that is nothing but as n tends
to infinity X n of w will tends to X of w
and collect those w's that is going to be
the event A. If the probability of that event
A is going to be 1; then, we can conclude
the X n converges to X almost surely.
In other words, negation of the event A that
probability is going to be 0; that means,
you collect all possible outcomes from omega
which does not converge X n of w to the X
of w, you collect those possible outcomes
whose measure is 0. Then, we also we can conclude
X n converges to X almost surely; that means,
it is the the whole unit mass is attached
with the collection of possible outcomes;
those outcomes or the outcomes of X n of w
converges to X of w.
Those outcome satisfies X n of w converges
to r tends to X of w as n tends to infinity.
So, if you include those possible outcomes
or w's those possible outcomes whose probability
mass is 1 or whichever is not satisfy in this
condition, those possible outcomes whose probability
mass is 0. Then we can conclude X n converges
to X in almost surely. We will go for the
example for this.
Let omega F and P be a probability space.
Let sequence of random variable defined on
the probability space omega [FL] P as X n
of X n is is mapping from omega to real line
such that X n of w is equal to 1 plus 1 by
n for n is equal to 1, 2 and so on.
Now, will go for computing limit n tends to
infinity X n of w ah where w is belonging
to make a limit; n tends to infinity of X
n of w that is going to be 1.
That means, the set of all w's belonging to
omega such that X n of w will tends to 1.
You collect those possible outcomes find out
the probability of that that is nothing but
P of omega that is same as 1. In that case,
one can conclude X n converges to 1 almost
surely.
Because for all because for all w belonging
to omega the limit n tends to infinity X n
of w is equal to 1. Therefore, finding the
probability of w belong into make a satisfying
this condition that is X n of w tends to 1
as n tends to infinity that is going to be
the whole possible outcomes that is nothing
but the p of omega that is equal to 1. Therefore,
it satisfies the condition for almost surely.
Therefore, X n converges to 1, almost surely.
So, the same example can be considered for
other mode of convergence also.
That means, one can prove limit n tends to
infinity. The probability of absolute of X
n minus 1 greater than epsilon that also we
can prove this is going to be 0. This implies
X n converges to 1 in probability.
Similarly, for the same example, one can prove
the sequence of random variable CDF's has
a limit n tends to infinity will converges
to F of X, where F of X is where F of X is
takes a value 0 till 1 and 1 onwards, it is
going to be 1. Therefore, we can conclude
the same example X n converges to 1 in distribution
.
That means, there are few a sequence of random
variable may converge to more than 1 mode
of convergence. So, that can be connected
in 1 nice way that is suppose you have a sequence
of random variable and the random variable
X defined on the probability space omega F
P convergence in distribution and we have
a convergence in probability .
Suppose, some sequence of random variable
convergence in probability this implies convergence
in distribution. The converse is not true
it may be true for some few problems, in generalities
not true.
Therefore, I am not writing the up adder converges
not true in general. Similarly, if you have
a X n converges to X in the first order moment,
this implies the converges in probability;
Converse is also not true in general here.
Suppose, you have a convergence in rth moment
that implies the r minus 1 and so on till
the first order moment.
But here, again converse is not true because
of 2 reasons. The first order moment exist
that does not being that the second order
moment is going to exist; even if it exist
it does not imply the convergence takes place.
Therefore, the converse is not true. Convergence
in a rth moment exist, then you can go for
tilde first order moment convergence that
implies the convergence in probability that
implies convergence in distribution.
Now, I am giving the connection with the convergence
in almost surely that is X n converges to
X almost surely. This implies convergence
in probability, you see the different direction.
Almost surely convergence implies convergence
in probability convergence in probability
implies convergence in distribution; no where
converse is true in general.
There are some additional condition we can
mention for some sequence of random variable,
then we can conclude the X n converges in
X in probability that convergence in almost
surely also. But for that you have to make
a some additional condition. In general, the
converse is not true for all the cases.
So, with this relation I am going to stop
the convergence or modes of convergence. In
the next class we will discuss about the law
of large numbers.
