Determine the mean and autocovariance function of xt, and state whether it is stationary

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I'm trying to understand the process to answer this question.

Let $w_t,$ for $t=0, \pm1, \pm2,...$ be a normal white noise process, and consider the series:

$x_t=w_tw_{t-1}$

Determine the mean and autocovariance function of $x_t$, and state whether it is stationary or not.

Can someone explain to me behind this logic:

$\gamma_x(h)=Cov(x_t,x_{t+h})=Cov(w_tw_{t-1},w_{t+h}w_{t+h-1})=\sigma^4_w $