Let's say we have a white noise process $x(t)$ such that:



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:


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.

  • $\begingroup$ better ask it on signal processing stack exchange ....there you'll get answer from signal point of view and more physical insight regarding White noise SSS(strict sense stationary) process ..it better suits there. $\endgroup$ – user454960 Dec 23 '18 at 10:44

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