Covariance for periodic weakly stationary process

Question

Let $X(n),n\in \mathbb N_0$ be a weakly stationary process with $X(n) = X(n+N)$ for some $N \in \mathbb N_0$.

What is the covariance function $b(k):=\operatorname{Cov}[X(n+k),X(n)]$?

Answer 1

The hypotheses are not enough to determine $b(\ )$. Consider for example $X(n)=(-1)^nX(0)$ with $X(0)$ symmetric, then $X(n+2)=X(n)$ for every $n$ and $b(k)=(-1)^k\mathrm{var}(X(0))$ for every $k$. But $X(n)=X(0)$ with $X(0)$ symmetric yields $X(n+2)=X(n)$ for every $n$ and $b(k)=\mathrm{var}(X(0))$ for every $k$.

Answer 2

If you introduce two random variables $X_1 = X(n)$ and $X_2 = X(n + N)$ then the covariance function reads $$b(n_1,n_2) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x_1 - \bar{X_1}) (x_2 - \bar{X_2}) f_{XX}(x_1,x_2) \,\mathrm{d}x_1\,\mathrm{d}x_2$$ then, assuming weak stationarity implies $\mu\lbrace X(t)\rbrace = \mu$ and $b(n_1,n_2) = b(N)$, thus $$b(N) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x_1 - \mu) (x_2 - \mu) f_{XX}(x_1,x_2) \,\mathrm{d}x_1\,\mathrm{d}x_2$$