Relationship between autocorrelation function and mean of random process For instance, if we assume a white noise process $w(t)$, it would have an autocorrelation function $R=\delta(t)$. Obviously, the white noise process would have zero mean, but how can the autocorrelation function be use to prove that?
 A: White noise is generally defined to be a zero-mean process whose autocorrelation function is $K\delta(t)$ where $K > 0$
and $\delta(t)$ denotes the
Dirac delta, and so there is nothing left to prove. Alternatively,
a white noise process is a weakly stationary process whose power
spectral density $S(f)$ has constant positive value $K$ for all $f$.
For any weakly stationary (also called wide-sense stationary)
random process whose autocorrelation function $R(t)$ is an ordinary
non-periodic real-valued even function with a peak at $t=0$ (e.g. $e^{-|t|}$) such that the limiting value of $R(t)$ as $t \to \infty$ exists, the limiting value is the square of the mean $\mu$ and the power spectral 
density $S(f)$ includes a Dirac delta at $f=0$. Such a power spectral
density does not match the power spectral density of white noise at all.
So, why can't this notion of $\mu^2 = \lim_{t \to \infty} R(t)$
be applied to the autocorrelation function $K\delta(t)$ of white
noise to deduce that white noise must have zero mean? Well,
$\delta(t)$ is not a function at all; it only walks like a duck
and quacks like a duck inside various integrals, not everywhere
like all actual ducks do. In particular, $\delta(5)$, say, does
not mean the value of the Dirac delta at $t=5$, and so the notion
$\lim_{t \to \infty} \delta(t)$ has no meaning.
