# Autocorrelation-properties

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

ππΏ = βπ₯π’π¦πββπΉπΏπΏ (π) Proof : Since {X(t)} is a stationary process, then as π β β,π(π‘)πππ π(π‘ + π) are independent and ππ = πΈ[π(π‘)] πππ(π) = πΈ[π(π‘ + π)π(π‘)] limπββπππ(π) = limπββ πΈ[π(π‘ + π)π(π‘)]

=limπββ πΈ[π(π‘ + π)]πΈ[π(π‘)] = limπββ ππππ = ππ^2

β ππ = β(limπββπππ (π)) β΄ πΈ[π(π‘)] = β(limπββπππ (π))

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.

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