Given two random processes A(t) and B(t), we can talk about their correlation
f(s)=⟨A(s)B(0)⟩,
with angle brackets denoting ensemble average.
It is common to classify f(s) based on its asymptotic behavior, like exponential or power decay. Are there other classifications of correlation functions in use, based on some mathematical properties? What are the respective physical usages/interpretations?
For example, I can think of correlation functions for which the following property holds
∫∞0sf(s)ds=O(1f(0)[∫∞0f(s)ds]2),
O is for the big-O notation, "order of". That formula basically means that ∫∞0sf(s)ds∼τf∫∞0f(s)ds, with τf being the correlation time.