I have a question about a weak stationary process. If the covariance function is not dependent on $|t-s|$ nor any value of $s,t$ then is it also a weakly stationary process? This question arose since I have $$ X_t = \theta Z_{t-1} Z_t + Z_t , \quad t = 1,2,...; Z_t \overset{iid}{\sim} N(0,\sigma^2) $$ and I found $\mathbb{E}(X_t) = 0$ and $Cov(X_t,X_s) = \sigma^2$ for $t = s$ and $Cov(X_t,X_s) = 0$ for $t \neq s$. Based on this I also wonder if the mean and covariance function of the process does not depend on $t,s$ so it is a weakly stationary process? I would appreciate some reference to go into some definitions, since in general I have seen that the mean should not depend on $t$ and the covariance function should depend on $|t-s|$. Greetings
1 Answers
I will assume that all random variables considered have finite second moment, as this is a requirement for the definitions. Then what you have defined is a so-called white noise process see here, i.e. any process $(Z_{t})_{t \in \mathbb{Z}}$ satisfying $\mathbb{E}(Z_{t}) = 0, \mathrm{var}(Z_{t}) = \sigma^{2}, \forall t\in\mathbb{Z}$ and $\mathrm{cov}(Z_{t}, Z_{s}) = 0, t \neq s$ is such a process (note: some authors also allow for $\mathbb{E}(Z_{t}) = \mu \in \mathbb{R}$).
A weak stationary process has to satisfy the same conditions on the mean, but the covariance conditions becomes $$ \mathrm{cov}(Z_{t}, Z_{s}) = \gamma(|t-s|). $$ So in particular, the process that you have defined also satisfies this property, because we can use the map $\gamma(0) = \sigma^{2}$ and $\gamma(k) = 0, k \geq 1$. In the definition, dependence on $|t-s|$ does not imply that the map $\gamma$ (called the autocovariance function) cannot be constant for $k \neq 0$, so therefore any white noise process is in fact weakly stationary.
In general, for square integrable random variables, we have the relation: $$ \mathrm{i.i.d.} \subset \text{White Noise} \subset \text{Weakly Stationary}. $$ You should be able to find this in any reference textbook on time series analysis.
One counterexample for where the above relationship fails would be an i.i.d. sequence of Cauchy random variables, because their mean and variance is undefined/not finite.
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