Our next example is a so-called moving average process. This process is very popular in the context of a econometric. This process actually means that Yt up is a linear combination of the variance of the whiteners process Xt from previous time points, that is Yt is equal to Xt plus a1 X t minus one plus so on, plus aq X t minus q. Here a1 and so on aq are real numbers. And for simplicity we'll assume there is also a coefficient here, a zero which is equal to one. So this is a very popular model and in the context of our course, it will be very natural to ask whether this process is tertiary honored. First of all, let me mention that mathematical expectation of Yt is actually equal to zero. This is a very simple corollary of the fact that the mathematical expectation of Xt is equal to zero. As for the covariance function, it is equal to covariance between the sum j from 0 to q aj X t minus j. And this sum k from zero to q ak X s minus k. Now, we can guess linearity of the covariance function again is equal to the sum, j from zero to q and sum k from zero to q of the aj ak, and here I should write covariance between x t minus j and x s minus k. You know that xt is a white noise and we have already calculated the covariance function of this was process. Therefore, I just used that result is equal to sigma squared multiplied by the indicator that t minus s is equal to j minus k. This formula means that the function kts can be represent as the function gamma at time point T minus s, because only the difference between the arguments play some role here. In particular, if we consider a partial case of this process which is mainly denoted by m a of q so there's q parameters. So the case when q is equal to one, we get this auto covariance function is not equal to zero, only three points. So it's some number multiplied by the indicator. It's an absolute value of X is equal to one plus some value multiplied by the indicators that X is equal to zero. So finally will conclude that the moving average process is a weakly stationary process and the function gamma auto covariance function can be written in closed form. Our last example will be the auto regressive model. It's denoted as AR of p. This model assumes there's the process Yt is defined as follows. So this is B1 Y t minus one and so on, Bp Y t minus p plus some white noise epsilon t. Normally, it is also assumed that the covariance between epsilon t and Ys is qual to zero for all t largrer than s. So this is also a very popular model in econometrics and this also would be very natural to ask whether this process is stationary or not. Mathematical expectation of Yt and other objects are not very clear from this representation, and actually it turns out that this equation has various solutions. And let me consider more precisely the case when p is equal to one. That this same model is Yt minus some B multiplied by Y t minus one is equal to epsilon t. Here b can be any real number. It turns out that even this equation has various solutions and one of the solutions is the following: so Yt is equal to the sum j from zero to infinity. B is a power J multiply it by epsilon t minus j. So this is a close form for one of the solutions. It's very simple to show that these Y t is exactly solution of this equation and it turns out that this solution is stationary under some very simple assumption. To show this assumption, let's mentions that methodical expectation of Yt is definitely equal to zero and this was a covariance function. We can use exactly the same ideas as in our previous example when we considered the moving average process and what we have here is double sum j k from 0 to infinity B to the power J plus K multiplied by sigma squared and multiplied by the indicator that t minus s is equal to g minus K and here, if you employ the same ideas as in the previous example, if you will take t minus s equal to some specific number. For instance, if we take t minus s equal to zero. That is, t is equal to s, then j should be equal to K, and this sum would be the sum of b to k. K from 0 to infinity. You know there this sum converges if and only if absolute value of b is smaller than one. And if you will think about the situation you will immediately realize that only in this case, the corresponding infinite sums converge. This is exactly is the answer. The absolute value of b is smaller than one, then definitely this process Yt is weakly stationary. In more general case when you have arbitrary p, you should consider some equation related to this coefficients and you should consider all complex roots of this equation, and if all of them lie inside the unit circle, then this process is stationary. I think the details can be found in any course on econometrics and I would like to skip them now. This was an introduction to the concept of stationarity. I hope very much that my examples helped you to understand this topic and it was found that even if a process Xt is weakly stationary, this property helps a lot to understand some other more complicated properties of Xt. I will discuss many further properties of stochastic processes later, but it will be very helpful for you to understand what does the notion of stochastic integration mean and in the next subsection, I will give you an intuition about this notion at least in simple situations.
