Stationarity of a Stochastic Process

Question

Let us say we have a Stochastic Process $X_t$ defined as follows: $$X_t= \sum_{i=0}^j \alpha_i\epsilon_{t-i} \,\, + \sum_{i=0}^k \beta_i\gamma_{t-i}$$ where $\epsilon$ and $\gamma$ are mutually independant, normally distributed white noise processes with finite variances $\sigma_\epsilon^2$ and $\sigma_\gamma^2$. (Also assuming that $\alpha$ and $\beta$ are non-zero and $j,k$ are positive.) How can one determine the stationarity of this stochastic process? Is it weakly stationary, strongly stationary or not stationary at all?

Answer 1

$E(X_t)$ is obviously zero. Because $\gamma$ and $\epsilon$ are iid, for $Var(X_t)$ we get $Var(X_t)= \sigma_{\epsilon}^2 \sum_{i=0}^j \alpha_i^2\ + \sigma_{\gamma}^2\sum_{i=0}^k \beta_i^2$.

$E((X_t -\mu)(X_{t-h} - \mu)) = E(X_t,X_{t-h})$, beacuse $\mu = 0$

So we have:

$$E((X_t -\mu)(X_{t-h} - \mu)) = E\bigg[\bigg(\sum_{i=0}^j \alpha_i\epsilon_{t-i} + \sum_{i=0}^k \beta_i\gamma_{t-i}\bigg)\bigg(\sum_{p=0}^j \alpha_i\epsilon_{t-p-h} + \sum_{p=0}^k \beta_i\gamma_{t-p-h}\bigg)= \\ =E\bigg[\bigg(\sum_{i=0}^j \alpha_i\epsilon_{t-i}\bigg)\bigg(\sum_{p=0}^j \alpha_p\epsilon_{t-p-h}\bigg)\bigg] + E\bigg[\bigg(\sum_{i=0}^k \beta_i\gamma_{t-i}\bigg)\bigg(\sum_{p=0}^k \beta_p\gamma_{t-p-h}\bigg)\bigg]$$,

because $E[\epsilon_t\gamma_s] = 0 \hspace{.2cm} \forall t,s \in \mathbb{N}$

If $h > k$, $E\bigg[\bigg(\sum_{i=0}^k \beta_i\gamma_{t-i}\bigg)\bigg(\sum_{p=0}^k \beta_p\gamma_{t-p-h}\bigg)\bigg] = 0$, beacuse $\gamma$ is i.i.d.

Analogously $h > j \Rightarrow E\bigg[\bigg(\sum_{i=0}^j \alpha_i\epsilon_{t-i}\bigg)\bigg(\sum_{p=0}^j \alpha_p\epsilon_{t-p-h}\bigg)\bigg] = 0$.

If $ h \leq j \Rightarrow E\bigg[\bigg(\sum_{i=0}^j \alpha_i\epsilon_{t-i}\bigg)\bigg(\sum_{p=0}^j \alpha_p\epsilon_{t-p-h}\bigg)\bigg] = E\bigg(\sum_{\substack{i=0\\p=0\\i=p+h}}^j \alpha_i\alpha_p\epsilon_{t-i}\epsilon_{t-p-h}\bigg) =\\ =\sigma^2_{\epsilon}\sum_{p=0}^{j-h}\alpha_{p+h}\alpha_p$

Similarly if $h \leq k E\bigg[\bigg(\sum_{i=0}^k \beta_i\gamma_{t-i}\bigg)\bigg(\sum_{p=0}^k \beta_p\gamma_{t-p-h}\bigg)\bigg] = \sigma^2_{\gamma}\sum_{p=0}^{k-h}\beta_{p+h}\beta_p$

Which implies that the autocovariance does not depend on $t$ and the time series is stationary.