I'm having trouble understanding part of one of the examples here. This is taken from Hamilton's book Time Series Analysis, p. 194.
My question is this. I don't understand why $$ E[X_t^2] = E[\epsilon^2 Y_{t-k}^2] = E[\epsilon_t^2] \cdot E[Y_{t-k}^2] $$ in example 7.15. I understand that proposition 7.10 tells us that $E[X_t^2]$ exists, but I don't understand why we can say $E[\epsilon_t^2 Y_{t-k}^2] = E[\epsilon_t^2] \cdot E[Y_{t-k}^2]$.
Thanks in advance.
Because $Y_{t-k}$ is measurable with respect to the family $(\epsilon_{t-k-j})_{j\geqslant0}$, which is independent of $\epsilon_t$ since $\{t\}\cap\{t-j-k\mid j\geqslant0\}=\varnothing$, and because this implies that $Y_{t-k}$ and $\epsilon_t$ are independent.