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Consider the following stochastic process: $$X_t = \epsilon_t\epsilon_{t-1},~~~~~~~~~\epsilon_t \sim N(0, \sigma^2)$$

Determine whether the process is covariance-stationary, strictly stationary, integrated of order one or neither of these.


I have some problems with the mean of this model:

$$E(X_t)=E(\epsilon_t\epsilon_{t-1})$$

The expectation is given by: $$E(X_t)=\gamma(1)$$ $$\text{or}$$ $$E(X_t)=E(\epsilon_t)E(\epsilon_{t-1})=0 ~~~???$$



If $\epsilon_t$ had been distributed as a $WN(0, \sigma^2)$, there would have been some differences in the expectation of $X_t$? or would the result have been the same?

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When noise is Gaussian and independent and identically distributed (i.i.d.)

$$ \mathbb{E}[\epsilon_t \epsilon_{t-1}] = \mathbb{E}[\epsilon_t] \mathbb{E}[\epsilon_{t-1}]= 0 $$

since $\epsilon_t$ and $\epsilon_{t-1}$ uncorrelated.

Also note that $\gamma(1)=\mathbb{E}[X_t X_{t-1}] = \mathbb{E}[\epsilon_t \epsilon_{t-1}\epsilon_{t-1} \epsilon_{t-2}] $ and that also happens to be equal to $$\mathbb{E}[\epsilon_t] \sigma^2 \mathbb{E}[\epsilon_{t-2}]=0$$

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