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I am trying to intuitively make sense of a specific property of the autocorrelation function: $$ \lim_{\vert\tau\vert \rightarrow \infty} R_{XX}(\tau) = \bar{X}^2 $$ where $\bar{X} = \mathbb{E}[X(t)]$ is the expected value of the random process $X(t)$. Here, $X(t)$ is ergodic in nature and has no periodic components. Can anyone help me prove this mathematically, without using the power density spectrum function?

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A very hand-waiving but still could be powerful way of visualizing this property could be to list that:

$$ \mu_x(t)=E[x(t) ]=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}x(t) dt=\widehat{\mu_x(t) }$$

for ergodic processes. And when the autocorrelation is concerned:

$$R_{xx}(\tau) =E[x(t) x(t+\tau)]\implies \lim_{\tau\to\infty}E[x(t) x(t+\tau)]=E[x(t)]E[x(\tau)]=\left(\widehat{\mu_x(t) }\right)^2$$

as tau goes to infinity, $x(t)$ and $x(\tau)$ tend to be independent hand-waivingly while we are not talking about periodic processes. And that simply becomes the square of the mean.

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