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We usually define the autocovariance function of a discrete-time weak stationary process as $\gamma(h) := \gamma(h,0) = \gamma(r-s,0) = \gamma(r, s) := \text{Cov}(X_r, X_s)$ with $r,s \in \mathbb{Z}$.

Is it possible that for some lags $h \in \mathbb{Z}$, $\gamma({h}) = \infty$ and can an example be provided? I know the variance is always finite ($\gamma(0) < \infty$) but what about for $h \ne 0$?

Darby Bond
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  • You link to a concept usually known as weak stationarity. That's not quite the same as a stationary process, but the difference is enough to call into question your answer. Please, then, clarify your definition. – whuber Oct 18 '20 at 15:41
  • okay, I'll change the name to weak stationarity. Do I need to add the definition for weak stationarity even if I have put up a link? – Darby Bond Oct 18 '20 at 15:45
  • You probably don't need to be that explicit, because the term is standard -- but because the answer hinges on the presumption of finite variance, you might wish to point out that the definition includes this assumption. – whuber Oct 18 '20 at 15:54
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    Agreed, no problem. – Darby Bond Oct 18 '20 at 15:55

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Actually, it is not possible. We know that for stationary processes, $|\gamma(h)| \leq \gamma(0)$, $\forall h \in \mathbb{Z}$ and since $\gamma(0) < \infty$ we are done.

Darby Bond
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    Consider the case of a stationary process of I.i.d. Cauchy random variables. – Dilip Sarwate Oct 18 '20 at 14:10
  • a sequence of Cauchy random variables cannot be stationary because each random variable has infinite raw second moment – Darby Bond Oct 18 '20 at 15:23
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    And where in the definition of what is meant by a stationary process do you find the restriction that all the random variables have finite second moment? All that stationarity means is that for all choices of positive integer $n$ and $n$ time instants $t_1, t_2, \ldots, t_n$, the joint distribution of $X(t_1), X(t_2), \ldots, X(t_n)$ is the same as the joint distribution of $X(t_1+\tau), X(t_2+\tau), \ldots, X(t_n+\tau)$, a condition that is easily satisfied by iid random variables regardless of what their common distribution is. – Dilip Sarwate Oct 18 '20 at 15:34
  • It's in my question, but I'll place it here – Darby Bond Oct 18 '20 at 15:35
  • Your link is to the definition of wide-sense-stationarity and not stationarity at all. Stationarity is defined earlier on the same page in Wikipedia, and their definition matches what I said in a previous comment. – Dilip Sarwate Oct 18 '20 at 22:53