Physical meaning of integral (infinite sum) of correlation function Title says it all.
What is a meaning of a integral (for continuous) or infinite sum (for discrete) of a correlation function?
I see many Fourier transformations and  convolutions but that is not what I'm looking for, although it. could be related.
 A: Consider the correlation function
$$
R(t_1,t_2) = \langle X(t_1)X(t_2)\rangle
$$
Suppose $X(t)$ is a stationary process. This implies that
$$
R(t_1,t_2) = R(t_1-t_2,0) = R(t_1-t_2)
$$
The Wiener-Khinchin theorem tells us that the autocorrelation function is the Fourier transform of the power spectral density $S_{XX}(f)$. The power spectral density tells us how much noise content a signal has a in a given frequency window. The Wiener-Khinchin theorem can be stated as:
\begin{align}
S_{XX}(f) = \int_{t=-\infty}^{\infty} e^{i 2\pi f t} R(t) dt
\end{align}
If we evaluate this at zero we see
$$
S_{XX}(0) = \int_{t=-\infty}^{+\infty} R(t)dt
$$
This is the answer to your question. In the case of stationary processes the integral of the auto correlation function gives you the power spectral density evaluted at $f=0$. The power spectral density at $f=0$ tells you about the DC component of your signal. For example, if the integral of the autocorrelation function is 0 this would tell you your signal has no DC component.
The answer for the discrete case is the same, you just work with discrete Fourier tansforms and discrete power spectral densities instead.
I don't have a ready interpretation for this integral if the signal is non-stationary.
