Assume we have a sequence of i.i.d normal random variables ${Z_t}$, with mean 0 and variance 3.
- Let $X_t = Z_tZ_{t+1}$
I understand that the expected value of $X_t$ can be written as $E[X_t]=E[Z_tZ_{t+1}]=E[Z_t]E[Z_{t+1}]$ because they are i.i.d variables. But how about $E[X_t^2]$?
$$E[X_t^2]=E[Z_t^2Z_{t+1}^2]$$
Since $Z_t$ is not independent of itself, can we write $E[Z_t^2Z_{t+1}^2]=E[Z_t^2]E[Z_{t+1}^2]$?
- Let $X_t = Z_tcos(t)+Z_{t+1}sin(t+1)$ where $Z_t$ is not independent of variable $t$.
How can we deal with $E[X_t^2]$ in this case?
For example, let's take one term from $E[X_t^2]$
$$E[Z_tZ_{t+1}cos(t)sin(t+1)]$$
can we further expand this term since $Z_t$ and $Z_{t+1}$ are i.i.d random variables?