On the notion of reasonable functions in Cohen--Lenstra type heuristics

Abstract

We elucidate the notion of "reasonable functions" in the context of Cohen--Lenstra heuristics. For a sequence $(x_i)_i$ of independent random variables valued in an infinite discrete probability space $(X,p)$, we investigate for what sets $S$ of functions on $X$ it is true that almost surely, for all $f\in S$, the average of $f$ on the sequence is equal to its expected value. We show that if $p$ decays sufficiently quickly, as e.g. the Cohen--Lenstra distribution does, then $\bigcup_{q>1}L^q(X)$ is such a set; while if $p$ decays sufficiently slowly, then not even $\bigcap_{q<\infty}L^q(X)$ has this property.