Square of stationary variable

Question

I'm given a stationary process $x_t$: $$\Bbb E(x_t) = \mu $$ $$Cov(x_t,x_{t+k}) = \xi_k \text{ some function of } k \text{ and not of } t$$ I need to find $ \Bbb E(x_t^2)$ using $\mu$ and $\xi_k$.

Answer 1

We can use the formula for variance:

$$ \Bbb Var(x_t) = \Bbb E(x_t^2) - \Bbb E^2(x_t)$$ From this follows that $$ \Bbb E(x_t^2) = \Bbb Var(x_t) + \Bbb E^2(x_t)$$

Knowing that the process is stationary, we have $$\Bbb E^2(x_t) = \mu^2$$ $$\Bbb Var(x_t) = \Bbb Cov(x_t, x_t) = \Bbb Cov(x_t, x_{t+0}) = \xi_0$$

Therefore, $$ \Bbb E(x_t^2) = \xi_0 + \mu^2$$