Throwing Stones and Collecting Bones: Looking for Poisson-like Random Measures

TL;DR

This paper characterizes three fundamental Poisson-like random measures—Poisson, binomial, and negative binomial—that remain self-similar under restriction to subspaces, with potential applications in various fields.

Contribution

It identifies and characterizes the only three rescaled self-similar random measures within a broad class, expanding understanding of their properties and applications.

Findings

Poisson, binomial, and negative binomial measures are the only self-similar measures under restriction.

Provides simple examples demonstrating applications of these measures.

Establishes a foundational classification of Poisson-like random measures.

Abstract

We show that in a broad class of random counting measures one may identify only three that are rescaled versions of themselves when restricted to a subspace. These are Poisson, binomial and negative binomial random measures. We provide some simple examples of possible applications of such measures.