Welcome back. We have discussed monotone convergence
theorem which says that if
you sequence of functions converging almost
everywhere monotonically. Then you can
exchange limit an integration write integral
of limit same as limit of integration. Today,
we will discuss dominated convergence theorem
which gives another sufficient condition
for you to interchange change limit and integration.
So, for dominated convergence
theorem essentially says that if you have
a sequence of functions which converges to
another function not necessarily monotonically.
But suppose that the sequence of
functions is bounded above by another function
which has a finite integral then you can
into change limit integration. So, that is
what is dominating convergence theorem says
are you the very useful theorem.
In particular if you have a sequence of bounded
functions you can always into
interchange limit an integration. Before the
dominating convergence theorem can be
derived as a consequence of monotony it is
like a corollary of the monotony convergence
theorem. So, in some sense the M C T is all
there is chew at alright and this for 2 Selma
is also corollary of monotone convergence
theorem in fact, the where you do it is from
M
C T you can prove for Fatou’s lemma and
from for Fatou’s lemma you can you can
prove dominated convergence theorem. So, this
dominated convergence theorem is
practically very useful. So, and for Fatou’s
lemma it is a little bit more technical. So,
I
will discuss it, but I will not hold you responsible
for it I will not consider it a par part of
your syllabus. Dominated convergence theorem
is very useful and important. So, let us
discuss for Fatou’s lemma first.
..
So, before so, let me just modify motivate
like this note that if you have if X comma
y
are and a variables. So, we always have 
expectation of min of X y there is not equal
to
min of expectation of X expectation of y it
is always true you see why it is true. So,
this
is actually elements so, expectation of the
minimum is less than or is equal to minimum
of the 2 expectations. So, this is true is
you can see fairly easily this is, because
minimum
of X comma y is less than or equal to X and
minimum less than or equal X comma y is
less than or equal to is also less than or
equal to y. So, if you take expectations you
have.
So, this is less no or equal to that so, if
put expectation here I should get that right
and
similarly for this. So, I put expectation
of a let I should get so, this which a both
the
statements are true. So, expectation of min
of X y must be less no or equal to that is
smaller of these 2 write both of these are
true.
So, this should be so, if you take this is
true also for n random variable for a this
is same
argument extends for n random variables you
put X 1 through X n as random variables.
The expectation of the minimum is there is
no equal to minimum of the expectations
always true for finite number random variable.
This is true see this for Fatou’s lama in
the same spirit expect it consider as a infinite
it is consider as a sequence of random
variables is not a finite collection of random
variable X 1 through X n, but consider a
sequence of random variables. Now, if you
have a sequence of random variables X 1
through X 1 X 2 dot that is no notion of minimum
if you have infinite collection of
numbers you cannot necessarily talk about
the minimum, because minimum may not be
.attain right you have to talk about infimum.
So, that is what Fatous lemma it is it is
a
very analytic results it is a very results
for the case of finite number of random variables
for Fatous lemma generalizes it to a sequence
of random variables.
.
So, Fatous lemma let X n b sequence of random
variables such that 
X n is greater than or
equal to y for all n and expectation of flu
it y is less than infinity. So, I am just
stating it
is for the case of random variables you can
state it for functions also there is no big
difference. I will state both Fatous lemma
and dominated convergence theorem for these
random variables. Then 
expectation of . X n is less than or equal
to
lim in expectation of X n. And if you replace
X n by minus X n if you have X n is a
sequence of random variables as that ditto
X n less than or equal to y for all n. And
so,
there are expectation of y less than infinity
then 
expectation of lim sup X n is greater no
or equal to lim sup expectation of X n. So,
this part is I mean this is not a separate
statement this is a say statement 2 is same
as statement number 1 with X n replace with
minus X n and y replace with minus y. So,
if you prove this you the prove this after
all
lim in of minus X n is same as lim minus lim
sup of X n. So, recall what. So, remember
what lim in is before we go on and prove this.
..
So, recall what is lim inf n tending to infinity
of a n sum sequence a n what is this
defined as equal this is as this is limit
n tending to infinity infimum m greater than
or
equal to n a m this is the definition of lim
inf I think we said this in the very first
lecture.
So, I think a gave a little bit lower be of
limit someone. So, this so, you look at the
sequence for m greater than or equal to n
a n a n plus 1 and. So, 1 and you look at
that
infimum and then you send n to infinity. So,
E sincere looking at inf m greater than or
equal to n your looking at a n a n plus 1
and so, 1 this you can show to be an increasing
non decreasing sequence is n so this limit
always exists. So, lim inf is always well
defined limit may not exists, but the lim
inf for always exists lim sup is defined the
same
thing same way with supremum.
So, now, what is lim inf n tending to infinity
of X n? For each omega you consider X n
of omega and X n of omega will be a sequence
of real numbers like that. And lim inf of
X n ids simply for every omega you consider
lim inf of X n of omega which is defined
exactly like this. So, this you can show in
fact, to be a random variable lim inf of a
sequence of random variable always a random
variable lim inf lim sup limit there all
random variables you can prove it. So, if
you take the expectation of the lim inf random
variable you always get something that is
smaller than or equal to first taking the
expectations and then taking the lim inf of
that sequence. First you have a taking lim
inf
of the random variable sequence then taking
expectations this is less no or equal to
taking the expectation first and then taking
lim inf exactly like this. Except now, you
.have is a infinite sequence so, you cannot
talk about minimum you have to talk about
infimum and so, this limit and this is a limit
infimum let so, it is the smallest limit point
of the sequence. So, the statement 
with me to the proof is the follow. So, if
I prove 1 avid
of proof 2. So, I will let me just prove 1.
So, fix n and then we have 
infimum k greater than or equal to n X k minus
y I am going
to consider that X k minus y is less not equal
to X m minus y for all m greater than or
equal to n. So, this y sees y is a random
variable with finite mean which has which
is like
a lower bound on all is X n s. So, often this
Fatou’s lemmas applied for non negative
random variables in which case you simply
have y equal to 0. For example, so, if you
have a sequence of nonnegative random variables
X n is bigger no equal to 0 and you
can apply. Let us a particular case of a Fatous
lemma that is often how it is apply or any
lower bound so, this y is acting like a lower
bound. So, this is y is acting like a lower
bunk and y has finite me. So, you have this
statement. So, y is. So, you are looking at
this
sequence for n plus 1 and so, 1 and you looking
at the infimum. So, that must be smaller
than X n minus y for any m greater than or
equal to n write this is very clear from the
definition of the infimum.
So, you take expectations writes now, you
take so, you have expectation of infimum k
greater than or equal to n X k minus y is
less no or equal to expectation of X m minus
y
for all m greater than or equal to n. I am
essentially doing this kind of a step I am
doing
expectation on both sides. So, this is true
for all m greater than or equal to n. So,
now, I
am going to make the equal ant of this step.
So, I am going to take infimum over all m
greater than or equal to n see this say there
is no m. This say it is an m and this is true
for
all m greater than or equal to n. So, if a
put an infimum here see if a put an infimum
over
m here nothing will happen there is no m here
I can put an infimum. So, I get so,
basically all that is less than or equal to
infimum over m greater than or equal to n
expectation of X m minus y. So, now, I take
so, now, I will send n to infinity is I send
n
to infinity is so, I am going to do so, this
is so, I have take in infimum here. So, now,
I
am going to send n to infinity this guide
infinity. So, if I send n to infinity what
happens
to this side? It will be what limit so, it
will be become lim inf.
..
So, this if you look at this box if a put
limit n tending to infinity inf m greater
than or
equal to n I will have so, look at that whole
thing as a m. This should become lim inf of
expectation of X n minus y. Now, so, take
limit n going to infinity so, let me continue
over there I have.
.
So, I will have limit n tending to infinity
expectation of inf k grater or equal to n
X k
minus y there is less no or equal to right
hand side will become lim inf right lim inf
of
expectation of X n minus y. 
See if you are getting bother by this y think
of y is 0. I mean
.mostly Fatous lemma applied with y as a constant
random variable in particular equal to
0. It does not have to be, but it you usually
its applied if you are bothered by y just
forget
this y assume that it does not exist I am
going to write it, but pretend that it does
not
exist. If you this, because it is harder little
bit of hard work keeping track all these
infimum. So, it had first so, it may be want
ignore the y. So, here I have so, on this
side I
have what I want write if you forget this
y for the moment I actually have what I want
on
the right hand side on the left hand side
I have something little bit different.
So, ideally what would I like to do? So, if
this limit were to jump inside the expectations
what would I have I will have what I want.
So, if I am expect establish for Fatous lemma
if I can jump the limit inside the expectation
now is that allowed we know. So, for you
know only 1 result under which that is allowed
monotonic convergence theorem. So, I
have to see if let a say if you look at that
whole thing some z n. So, this is we index
by n
is not it this is index by n, because I am
taking infimum k greater than or equal to
n.
Now, this z n will be a increasing or decreasing
z n is non decreasing first of all z n is
not
0 now well non negative, because X n is greater
than or equal to y. So, this whole thing
is non-negative so, infimum of non-negative
sequence. So, z n is definitely non negative
for all n and your taking infimum k there
is no equal to n.
So, if you take if you increase n by 1 your
taking infimum over a smaller set at which
means it should be bigger is not it. So, and
z n is not decreasing so, note. So, this guides
z n and notes z n is non negative and z n
is non decreasing. So, that much is a very
easy
argument. Now, what is? So, ands limit so,
z n is a non-decreasing non negative
sequence of random variables. So, limit n
tending to infinity z n is we will simply
be lim
inf of what X n minus y. So, limit n tending
infinity of z n will be lim inf of X n minus
y
by definition of lim inf. So, now, so, I have
a sequence of random variables which
monotonically convergence to that random variable.
So, this is some other random
variable call this some z or something. So,
by M C T I can take the limit inside so, M
C
T gives the required result. So, m we have
apply M C T on the left side, because z n
is
monotonic and it convergence to some z I can
always take the limit inside and what will
have inside is z which is this guy. So, I
will have so, I mean it will give the required
result assuming y equal to 0.
So, if y is not 0 you will have something
like mc t gives let me write down will have
expectation of lim inf X n minus y it does
not equal to lim inf expectation of X n minus
.y. Now, if this if this y where is 0 no problem
if y is some random variable which has a
finite mean. Then I can invoke linearity of
expectations then expectation of that minus
that will be you know you will be like expectation
of this bit minus expectation of y and
similarly this expectation will go inside.
And that is where use this fact were expectation
of y is something finite alright so, you can
take the expectation inside and cancel the
expectation of y. But if this were not true
then you cannot this may not be well defined
write so, you with may have some infinity
minus infinity form so; if this were not true
then it does not work.
So, then linearity well help you prove it.
And the second part we follow if you apply
the
first part to the sequence minus X n and minus
y. So, this is not a distinct statement like
the same thing with X n replace with minus
X n and y replace with minus y, because lim
inf of X n will be minus lim sup of minus
X n any questions? So, Fatou’s lemma as
a
corollary of monotonic convergence theorem
see only nontrivial step see everywhere I
will just be in applying definition of lim
inf. And the only non-trivial step was in
applying monotone I in realizing that monotonic
convergence theorem can be use to take
the limit inside where that is only non-trivial
step. So, one of the main use of Fatous
lemma is not proving dominated convergence
theorem. It may also be of independent use
and some circumstances, but the dominated
convergence theorem follows very easily
from fatuous lemma.
.
.That is probably why fatuous lemma is called
a lemma may probably it is the lemma for
d c d you often 1 other earlier lemmas. In
mathematic you often wonder what it is a
lemma for there are any of this fundamental
results this is even lemma. I do not quit
know why it is called a lemma. It is probably
because its helps to the strong law of large
numbers it. So, that is why its call a lemma
I think dominate convergence theorem we
will use for Fatous lemma the as the lemma
for it we will use for Fatous lemma to prove
it. Consider a sequence X n of random variables
I will state it for random variables you
can state for functions also consider sequence
of random variables such that 
X n
convergence to X all most surely which means
on a set of probability 1 this happens.
Suppose there exists a random variable y such
that X n is less than or equal to y were it
absolute value of X n is not equal to y almost
surely and y has a finite expectation then
limit n dent infinity of X n is equal to expectation
of X.
So, you have a sequence of random variables
which X n as that X n convergence to X
almost surely, but this convergence need not
be monotonic you can be non-monotonic
also you can have X n bigger X n plus 1 smaller
you can have oscillations no problem.
But so, if that is all you had its not true
that you can interchange limit and expectation.
You know that if X n convergence takes almost
surely it is not necessarily the case that
the sequence of expectations will converge
to expect that is not the case. So, here you
have situation where X n are dominated by
another random variable y. So, absolute value
of X n is less no equal to y and y is a random
variable it is a finite mean finite
expectation then you can interchange limit
and expectation. So, limit of the expectation
is equal expectation of the limit random variable.
And the dominated convergence
theorem often you apply it for bounded random
variables.
So, if y often you if you can find some bound
on see if the X n s of for example uniform
random variables or something with which is
bounded almost surely which sorry for you
have a sequence of random variables converging
almost surely to some other random
variable. But this random variables are almost
surely bounded above by some random
variable y which has finite mean then you
can interchange limit and expectation. So,
let
us prove this 
since minus y is less than or equal to X n
is less than or equal to y for all n.
We can invoke both sides of Fatous lemma see
1 side of the side the side I prove says X
n is bigger than or equal to some random variable
with a finite mean the other side the
supremum side says is bounded above by some
random variable. Now, it is a say assume
.that absolute takes an as bounded above then
i necessarily have that. So, I can invoke
both sides of Fatous lemma. So, what you do
is the following its actually very simple
proof.
.
If you have expectation of X see expectation
of X is equal to the expectation of lim inf
of
X n. why this true? See, because limit of
X n is see the almost sure limit of X n is
X
alright since the almost sure limit of X n
exist then lim inf and lim super equal. So,
lim
inf of X n is same as the lim inf of X n that
we know that the limit exist rate almost sure
convergence happens so, you have this now.
So, as soon as you see this, what you do
invoke Fatou. Fatou says this is less no or
equal to lim inf expectation of X n thus so,
lim
inf expectation of X n always exists. So,
this is a well defined now, but lim inf is
less no
or equal to lim sup always. So, I have to
bring other side of fatou next. So, I have
to I
will do this less than or equal to lim sup.
So, this is, because a fatou part 1 right
and then
this is limit lim sup tend ending to infinity.
This is simply, because of the fact that lim
sup lees or equal to lim inf less than o E
equal to lim sup. Now, you can invoke the
other
side of fatou right saying that this is less
than or equal to expectation of lim sup of
X n
right this is, because of fatou number 2 part
2 of Fatous lemma.
Now, what is that equal to? It is just a expectation
of X again, because lim sup X n is
equal to limit of X n almost sure element
and because their the expectation of would
say
the almost sure is I am in on a set of problem
I think matters. So, this is equal to
.expectation of X. So, I only prove that expectation
or X is equal to expectation of X rate
actually I have the prove the less than or
equal to expectation v of X which I have
already new. I have proved this is less no
or equal to same thing which is always true.
So, why am a doing all this 
have a proved anything nontrivial here see
we just look at
this. And this it is looks like I have proved
it otology right proved a very trivial
statement, but I have proved something nontrivial
I have prove that all these inequality’s
must be equalities. If any of this so, there
are how many inequalities here 1 2 3
inequalities if any of these inequalities
were a strict inequality. Then I will get
a
contradiction saying expectation of X is strictly
less than expectation of X does not true
so, all these inequalities must be equalities.
So, in particular what is the non-trivial
quality here? So, this equality this lim inf
equal to lim sup is a consequence which means
the limit exists limit expectation of n X
n exists also limit expectation of X n will
be
equal to.
.
So, the, this implies so all equalities all
inequalities above must be met with equal.
Thus
lim inf of X n expectation of X n is equal
to lim sup expectation of X n equal to
expectation of X and since lim inf equal to
lim sup limit of a expectation or X n equal
to
expectation X. Therefore this is actually
a something on trivial here it is not it is
a he just
look at this and this you would looks like
even you not prove on anything. But it actually
proves that this and this must be equal which
means the limit has to exist on the limit
must be equal to expectation X. So, this dominating
random variable y just helps you to
.invoke Fatou’s lemma from both that X n
s that is only job of the y at as long as
you find
some random variable y which dominates absolute
X n and as long as.
So, it should have finite mean off course,
because the Fatous itself is not hold if there
it
is not true. Then you can always interchange
lim and expectation . 
any
questions? So, as corollary of 
dominated 
convergence theorem we well rather the most
common application of the dominated convergence
theorem is when y is some constant
m. So, if you have a sequence of random variables
which are all dominated by a constant
rate if there all bounded random variables
for example, then you can always interchange
limit and integration some people call that
corollary bounded convergence theorem. So,
now, we have a at least 2 situations where
you can interchange limit an integration M
C
T and d c t.
And if you think about it now, that you see
the proof d c t is simpler corollary of M
C T
rate them it is not really a distinct theorem.
It is stated as a separate theorem, because
it is
very practically very useful so, the M C T
is. So, all there is where this only 1 major
theorem in convergence of integrals so, let
us M C T any questions? Expectation or less
than infinity .. It is easy it is different;
because you need Fatous see
Fatous lemma need this kind of situation so,
you need X n to be. So, the Fatous lemma
part 1 said X n is greater than or equal to
y part 2 said X n is less than or equal to
some y.
So, without this you cannot invoke Fatous
lemma any other question?
Expectation of X less than infinity . you
cannot say that,
because even in the example we gave.
..
So, actually we gave this example if you remember
the function was equal to so, this is
my omega. So, 0 1 was my sample space and
my X n omega was so, in this I think so, it
what n and 1 over n remember this example.
So, in this case expectation of X n is always
1, but you cannot interchange expectation
and limit. In this particular case the sequence
of random variable X n expectation of X n
is finite for all n, but the X n are themselves
not dominated, because they are getting bigger
and bigger that is no random variable y
which dominates X n. So, is that expectation
of y is finite that were the case this may
will not have this is a counter example then
you will be a able to interchange limit and
expectation. So, this example neither monotonicity
nor dominated if, because either of
them holds you will have convergence of integrals.
It is having an expectation in that
example I will stop here.
.
