# $L^2$-stationary but not $L^2$-continuous process

I have to give a example of $L^2$-stationary (or also weakly stationary) but not $L^2$-continuous process.

By definition, for a $X(.)$ $L^2$-stationary process, $EX(t):=m(t)=c$, for all $t\in R$ and some constant $c$ and the covariance function depends only in the difference, i.e., $K(s,t)=K(0,t-s):=r(t-s)$.

We know that if $r:R \to R$ is continuous, then $X(.)$ is $L^2$-continuous. So, I'm looking for a $L^2$-stationary process with $r$ discontinuous.

So, I suspect there is a way to construct a simple random variable with these features. But I don't know it. I appreciate if you could give a hint.

How about Gaussian white noise? That is, $r(t-s)=0$ when $s\neq t$ but $r(0)=1$. – Byron Schmuland Jun 21 '12 at 23:26