Invariant expectation values in the sampling of discrete frequency distributions

TL;DR

This paper introduces invariant moments in sampling discrete frequency distributions, which remain constant across sample sizes, aiding analysis of scale-free systems and their limiting behaviors.

Contribution

It constructs a set of invariant moments with expectation values independent of sample size and applies these to scale-free distributions like Ewens sampling formula.

Findings

Invariant moments are independent of sample size.

Invariant moments can be computed for scale-free distributions.

Conditions for the existence of scaling limits are described.

Abstract

The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is discussed. A wide set of measurable quantities ("invariant moments") whose expectation value does not in general depend on the size of the sample is constructed and illustrated by applying the results to Ewens sampling formula. Invariant moments are especially useful in the sampling of systems characterized by the absence of an intrinsic scale. Distribution functions that may parametrize the samples of scale-free distributions are considered and their invariant expectation values are computed. The conditions under which the scaling limit of such distributions may exist are described.