On sampling and parametrization of discrete frequency distributions

TL;DR

This paper explores the relationship between discrete frequency distributions and their sample expectations, identifying invariant distribution forms and conditions for their scaling limits.

Contribution

It introduces a set of invariant expectation value combinations and characterizes distributions with sample expectation distributions that are form-invariant.

Findings

Identified combinations of expectation values independent of sample size

Found distribution functions with invariant sample expectation distributions

Described conditions for the existence of scaling limits

Abstract

The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is esablished. A set of combinations of expectation values whose value does not in general depend on the size of the sample is constructed. Distribution functions such that the distribution of the expectation values of their samples is invariant in form are found and studied. The conditions under which the scaling limit of such distributions may exist are described.