Hey guys, first time taking a time series class, need some help!

Let {$\displaystyle Z_{t}$} be an IID sequence with mean 0 and variance $\displaystyle \sigma^2$. (I think this is White Noise)

Let {$\displaystyle Y_{t}$} be a stationary sequence with a covariance function $\displaystyle \gamma_{y}(k)$. Then it says assume $\displaystyle Z_{t}$ and $\displaystyle Y_{t}$ are independent of each other.

Define $\displaystyle X_{t}=Z_{t}Y_{t}$.

Verify that for k ≥ 1 we have Cov($\displaystyle X_{t}$,$\displaystyle X_{t+k}$) = 0 and Cov($\displaystyle X^2_{t}$,$\displaystyle X^2_{t+k}$) ≠ 0, that is {$\displaystyle X_{t}$} is a white noise but not IID.

So Im not sure how to define $\displaystyle X_{t}$? Its the product of white noise ($\displaystyle Z_{t}$) and some stationary sequence ($\displaystyle Y_{t}$)?

Any help would be appreciated.