A Raikov-Type Theorem for Radial Poisson Distributions: A Proof of Kingman's Conjecture

TL;DR

This paper proves Kingman's long-standing conjecture that if the sum of two independent radial variables is radial Poisson, then each variable must also be radial Poisson, extending Raikov's theorem to radial distributions.

Contribution

It provides a proof of Kingman's conjecture, establishing a Raikov-type theorem for radial Poisson distributions, which was previously unproven.

Findings

Proves Kingman's conjecture on radial Poisson variables

Extends Raikov's theorem to spherical symmetry context

Clarifies the structure of radial Poisson sums

Abstract

In the present paper we prove the following conjecture in Kingman, J.F.C., Random walks with spherical symmetry, Acta Math.,109, (1963), 11-53. concerning a famous Raikov's theorem of decomposition of Poisson random variables: "If a radial sum of two independent random variables X and Y is radial Poisson, then each of them must be radial Poisson."