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How do we go about proving the property that entails; If X(t) is ergodic with no periodic components the autocorrelation converges to square of the mean as the time difference(Ο„) approaches infinity

𝝁𝑿 = √π₯π’π¦π‰β†’βˆžπ‘Ήπ‘Ώπ‘Ώ (𝝉) Proof : Since {X(t)} is a stationary process, then as 𝜏 β†’ ∞,𝑋(𝑑)π‘Žπ‘›π‘‘ 𝑋(𝑑 + 𝜏) are independent and πœ‡π‘‹ = 𝐸[𝑋(𝑑)] 𝑅𝑋𝑋(𝜏) = 𝐸[𝑋(𝑑 + 𝜏)𝑋(𝑑)] limπœβ†’βˆžπ‘…π‘‹π‘‹(𝜏) = limπœβ†’βˆž 𝐸[𝑋(𝑑 + 𝜏)𝑋(𝑑)]

=limπœβ†’βˆž 𝐸[𝑋(𝑑 + 𝜏)]𝐸[𝑋(𝑑)] = limπœβ†’βˆž πœ‡π‘‹πœ‡π‘‹ = πœ‡π‘‹^2

β‡’ πœ‡π‘‹ = √(limπœβ†’βˆžπ‘…π‘‹π‘‹ (𝜏)) ∴ 𝐸[𝑋(𝑑)] = √(limπœβ†’βˆžπ‘…π‘‹π‘‹ (𝜏))

This is how I proved the property, however I thought there was a more rigorous approach.

I would also like to know how we would work with the converse ( the presence of periodic components) when given only the autocorrelation.

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  • $\begingroup$ Take a look at these two related posts: 1 and 2. $\endgroup$
    – Matt L.
    Oct 25, 2023 at 11:11
  • $\begingroup$ Thanks, I have however my question is still not answered thus the reason I posted the question. $\endgroup$
    – Thabile
    Oct 25, 2023 at 15:42

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