The following proposition holds, let XT be stochastic process for discrete time and assumes as a co-variance function of this process is bound and therefore there is this sum constant alpha such that episode 12 as a co-variance function is less than alpha for any time moments 10 S. By the way it is completed as the same as to say as a variance of the process XT is bounded for any T. I don't want to show this is like a simple exercise, and let me also introduce the following notation by C of capital T I will know the co-variance between XT and the average value between the various X one and so on XT. Let me denote as average value by M capital T and MT is formally a sum T from one to capital T, XT and then we divide the sum by capital T. And this proposition tells us the following, that variance of MT converges to zero as T converges to infinity if and only if the value C of T is its co-variance between XT and MT converges to zero as T converges to infinity. I don't want to prove the statement. This is rather technical result, but let me show how we can apply it for checking the characteristic of a stationary stochastic processes. There's the following coloring will be at most important. Let me assume now that the protest XT is a weakly stationary. And I will denote its outer co-variance function by gamma, then the following two statements hold. The first statement. If one divided by capital T sum gamma of R from zero to T minus one gives this object converges to zero as capital T tends to infinity, then the protest XT is ergodic. And second item, if gamma of R itself diverges to zero as R tends to infinity, then the conclusion is the same, XT is ergodic. So, this coloring gives us two sufficient conditions which guarantees that the protest is ergodic. And of course we can check either the first condition or the second one. Now I would like to prove these two statements. Let me prove the first item. So, if the protest XT is ergodic, then the mathematical expectation is equal to a constant. And here we are immediately conclude the mathematical expectation of MT is also equal to the same constant. In this case what we have is that the variance of MT is actually equal to the mathematical expectation of MT minus this constant squared. And therefore, if we can prove that this mathematical expectation converges to zero, of course it is not proven at the moment, but if we can prove this, we can conclude that MT converges to this constant in the mean squared sense. And from here we conclude that MT converges to the same constant in probability sense. And therefore, the protest XT is ergodic. 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. So, what we have here, this function gamma of R is tending to zero, as R is tending to infinity. Moreover, it is equal to zero for a sufficiently large R. And therefore, due to the second part of this [inaudible] , we immediately conclude that the process XT is actually an ergodic process. Okay, let me now provide some further examples. Example is the following. Let me consider the process XT in the following form. It's A cosine Omega T plus B sine Omega T. Here A and B are two random variables which are uncorrelated. They can be dependent, but they're uncorrelated and they can also have different distributions. But let me assume for simplicity that mathematical expectations of A and B are equal to zero and variances of A and B are equal to one. As for the Omega, I would like to assume that is equal to some specific value, for instance, pi divided by 20. What we have here is a mathematical expectation of XT is equal to zero. This is just because of our assumptions and mathematical expectations of A and B equal to zero. And the covariance function is actually equal to cosine Omega T minus S. Well, from here it immediately follows that the process XT is weakly stationary. And as for the ergodicity, we shall check whether this or these assumptions are fulfilled. First of all, know that the function Gamma of R, which is in this case equal to cosine Omega R, is not tending to zero when R is tending to infinity. As for the first assumption, we claims that one divided by T sum cosine Omega R, R from zero to T minus one is less or equal than 10 divided by T. This is due to the same argument that in one of my first examples, this is just because cosine is a periodic function with period two pi. Well, until we finally get here that this sum converges to zero as T tends to infinity. So, we immediately conclude that XT is an ergodic process. And we'll have one more example of stationary ergodic processes. So we have now the examples of stationary processes such as the second condition is fulfilled and such as the first one is fulfilled. I guess, that the complete picture of this topic is now clear. Some of the process is not stationary. We shall check the ergodicity by some heuristic arguments. And if it is stationary, we have a couple of sufficient conditions which in most situations give us the conclusions that the process is ergodic. I guess this is all that you should know about ergodicity at the moment. Let me proceed to the next topic.