I am a little bit confused with the next example given in Brockwell and Davis,Time Series Theory and Methods, page 98. enter image description here

In the example, is computed the autocorrelation, and at some point, the author says that $Corr(Z_2,X_1)=0$ I don´t have clear why, because If I see the expression $$X_t=0.9X_{t-1}+Z_t$$ clearly $X_t$ depends on $Z_t$. Moreover If I write in particular $Z_2=X_2-0.9X_1$ evidently $Z_2$ is depending on $X_1$, how the correlation can be 0 ??


Note that you have written correlation at equal times ${\rm Corr}(Z_t,X_t)$, and the authors do not calculate this. They calculate ${\rm Corr}(Z_2, X_1)$, i.e. the correlation of the value of the random variable $Z$ at time $t=2$ with the value of the random variable $X$ at the previous moment in time $t=1$. These are independent by assumption, because this is how this stochastic process is generated: given the value of $X_t$ the random variable $X_{t+1}$ is generated using $Z_{t+1}$, which is assumed to have distribution independent of $X_t$.

What about $X_t$ and $Z_t$, i.e. values at equal times? As you have correctly noticed these are indeed dependent

$${\rm Corr}(Z_t,X_t) = {\rm Corr}(Z_t,0.9 X_{t-1} + Z_t) = {\rm Corr}(Z_t,Z_t)$$

In stochastic processes it is important to remember the time, at which you calculate something. It is like that at each moment in time $t$ you have a new random variable $X_t$ and even though it has the same name $X$ its properties are (related, but) different from those of $X_{t-1}$.


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