Why does $C(s,t)=\min(s,t)$ mean it is not the wide - sense stationary?

Question

I saw some information about Wiener process on the internet,and I can't really understand what it means. It says:

The covariance of Wiener process is $C(s,t)=\min(s,t)$ when $0\lt s \lt t$, so the Wiener process is not the wide-sense stationary(WSS)

Why is that a process isn't a WSS when $C(,) = \min(\,,\,)$?

Answer 1

For the process $Y_t$ to be WSS the following conditions must hold:

• $E(Y_t)=\mu_Y$ is not a function of $t$ (that is, has the same value for every $t$),
• $Var(Y_t)=\sigma^2_Y$ is the same for every $t$,
• $C(Y_s,Y_t)=\gamma_{|s-t|}$ depends only on the distance $|s-t|$ between $s$ and $t$ (that means—for instance—that $C(Y_2,Y_4)=C(Y_4,Y_2)=C(Y_7,Y_5)$, and so on).

In your example $C(Y_1,Y_2)=\min\{1,2\}=1$ and $C(Y_2,Y_3)=\min\{2,3\}=2$, and so these two covariances differ even though in both cases $|s-t|=1$.