Let's say we have a white noise process $x(t)$ such that:
$E(X(t)X(t+\tau))=N\delta(\tau)$
$E(X(t))=0$
In particular, with $\tau=0$, $E(X(t)X(t))=E(X^2(t))$ is infinite.
Now, I want $X(t)$ at each time $t$ to have a normal distribution of 0 mean and $\sigma^2$ variance. That is:
$E(X^2(t))=\sigma^2$
This is not consistent. I guess the expectations mean something different in both cases, but I don't find an explanation. This prevents me from moving ahead in a study of the mean, variance and autocorrelation of a process $y(t)$ defined as $y(t)=1$ if $a<x(t)<b$ and 0 otherwise.