What are the known classes of correlation functions?

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Given two random processes  $A(t)$ and $B(t)$, we can talk about their correlation

$$f(s) = \langle A(s)\, B(0) \rangle,$$

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

$$\int_0^\infty s\,f(s)\, ds = O\biggl( \frac{1}{f(0)} \biggl[ \int_0^\infty f(s)\, ds \biggr]^2 \biggr),$$

$O$ is for the big-O notation, "order of". That formula basically means that $\int_0^\infty s\,f(s)\, ds \sim \tau_f \int_0^\infty f(s)\, ds$, with $\tau_f$ being the correlation time.

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