Here, we essentially use the relation between different types of conversions.

You know that if some sequence converges to something in the two norms

that it also converges to the same random variable in probability.

Okay. The only question which remains,

is why variance from T is tending to zero.

But this is nothing more than the application of this theorem.

We should just check that the function capital C of

T is tending to zero when capital T tends to infinity.

Let us check this. So, capital C of T is equal to the co-variance between XT

and sum one divided by T sum XTT from one to capital T. So,

co-variance is a linear function,

therefore we can move one divided by T outside co-variance and

also we can write this co-variance as a sum of co-variances.

So, what we finally get is one divided by T,

sum T from one to capital T and here we have co-variance between X capital T and

X small T. But this is not similar as

the gamma function as a point capital T minus small T,

so is one divided by T sum T from one to capital T.

Gamma T minus T. And if we change the variable of summation here we

get nothing more than one divided by T sum R from zero to T

minus one gamma of R. Now let us complete this proof.

So, from the assumption of this corollary we have that this sum

is tending to zero as capital T tends to infinity.

Therefore, from this fact we conclude that variance of form T is tending to zero due

to this proposition and due to

this line of reasoning we immediately guess that XT is a ergodic protest.

This observation completes the proof.

And now I would like to show the second item of this corollary and the idea

here is quite simple just to show that if

gamma of R is such that this condition is fulfilled,

so the condition of the first item is also fulfilled,

and therefore I immediately get that the process XT is ergodic.

Let me recall one fact from calculus

the so called Stolz-Cesaro theorem.

The theorem tells us the following.

So, if you have two sequences of real numbers AN and DN,

and BN is such as that it is strictly increasing and unbounded.

And you know that limit of

AN minus AN minus one divided by BN minus BN minus one,

limit as N tend to infinity,

is equal to some number Q.

Then, due to the Stolz-Cesaro theorem,

we get that AN divided by AN also converges to the same constant Q,

when N tends to infinity.

These are rather interesting fact,

and this will help a lot to show that these fact,

that gamma R tending to zero guarantees this fact that the sum is also tending to zero.

Let us apply this Stolz-Cesaro theorem with appropriate choice of AN and BN.

More precisely, we will take AN equal to the sum R

from zero to N minus one gamma of R. And for BN,

we will take BN equal to N. Of course,

all conditions on the sequence BN are fulfilled.

This is too increasing and unbounded sequence.

And what we have here,

is the difference AN minus AN minus one divided by the difference BN minus

BN minus one is equal to gamma,

the point N minus one divided by one.

And according to the assumption of the second item of

[inaudible] we get the gamma of R tending to zero.

Therefore, this convergence is also tending to zero.

So, we have, in the Stolz-Cesaro theorem,

this Q is equal to zero.

And applying this theorem,

we immediately get that AN divided by BN,

namely one divided by N sum gamma of R,

R from zero to N minus one is also tending to zero,

to the same Q, as N is tending to infinity.

And therefore, this condition guarantees that this condition is also fulfilled.

And this would have shown already

that this condition guarantees that the XT is are ergodic.

We immediately conclude that from here it follows that the process XT is ergodic.

This observation concludes the proof and let me now

show how we can apply this [inaudible] in some situations.

Let me provide a couple of examples.

That NT be a Poisson process with intensity Lambda.

Of course, NT is not a stationary process just because

this mathematical expectation is not equal to a constant.

But if I now fix some constant P and define

the process XT as a difference between NT plus P minus NT.

This process has a mathematical expectation equal to mathematical expectation of

NT plus P that is Lambda multiplied by T plus P minus mathematical expectation of

NT minus Lambda T. These two terms vanish.

And we have that the mathematical expectation is equal to Lambda multiplied by P.

As for the covariance function,

it isn't a difficult exercise to show that it is equal to gamma T minus S,

where the function gamma of R is equal to

Lambda multiplied by P minus absolute value of R,

if absolute value of R is less or equal than P,

then it is equal to zero otherwise.