Suppose we have a stochastic process $x(t)$. Define the autocorrelation function (ACF) of this signal as
$$ G(t)= \lim_{T\rightarrow \infty} \int^T_0 \langle x(\tau)x(\tau+t)\rangle d\tau $$ My question is: what is the significance of this function evaluated at $t=0$?
My attempts at understanding are below. I understand that setting $t=0$ gives:
$$ G(0)= \lim_{T\rightarrow \infty} \int^T_0 \langle x(\tau)^2\rangle d\tau $$ which looks similar to the variance. Is it the case that $G(0) =\lim_{t\rightarrow\infty}\langle x(t)^2\rangle$ ? ie. Does $G(0)$ equal the steady state variance of the process?
As an example, I looked at the Ornstein-Uhlenbeck process, with $\dot{x}(t) = - a x(t) + D\xi(t)$ where $\xi(t)$ is a Wiener Process. Finding the time-dependent variance of this process gives: $$ \langle x(t)^2 \rangle = e^{-2at}\langle x(0)^2 \rangle + \frac{D^2}{2a}(1-e^{-2 a t}) $$
For this process, we also get an ACF of: $$ G(t) = \frac{D^2}{2a}e^{-at} $$ I notice that, when we take $t\rightarrow 0$, in the ACT this gives the steady state variance of the process $\lim_{t\rightarrow\infty}\langle x(t)^2\rangle$. Is this just a coincidence, or does this apply in general?