# Mean of $X_t = \epsilon_t\epsilon_{t-1}$

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?

$$\mathbb{E}[\epsilon_t \epsilon_{t-1}] = \mathbb{E}[\epsilon_t] \mathbb{E}[\epsilon_{t-1}]= 0$$
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 $$\mathbb{E}[\epsilon_t] \sigma^2 \mathbb{E}[\epsilon_{t-2}]=0$$