An Analogue of the L\'Evy-Cram\'Er Theorem for Multi-Dimensional Rayleigh Distributions

TL;DR

This paper extends the Lévy-Cramér theorem to multi-dimensional Rayleigh distributions by embedding Cartesian products of Kingman convolutions into symmetric convolutions, providing new theoretical insights into their structure.

Contribution

It introduces an embedding of multi-dimensional Kingman convolutions into symmetric convolutions and establishes an analogue of the Lévy-Cramér theorem for Rayleigh distributions.

Findings

Embedding of k-dimensional Kingman convolutions into symmetric convolutions

Analogue of the Lévy-Cramér theorem for multi-dimensional Rayleigh distributions

Theoretical framework for multi-dimensional Rayleigh distribution analysis

Abstract

In the present paper we prove that every k-dimensional Cartesian product of Kingman convolutions can be embedded into a k-dimensional symmetric convolution (k=1, 2, ...) and obtain an analogue of the Cram\'er-L\'evy theorem for multi-dimensional Rayleigh distributions.