Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean

TL;DR

This paper investigates the conditions under which the squared components of a Gaussian vector with non-zero mean are infinitely divisible, revealing a critical point beyond which infinite divisibility no longer holds.

Contribution

It establishes a precise criterion for infinite divisibility of squared Gaussian vectors with mean shifts and identifies a critical point where this property ceases.

Findings

Infinite divisibility holds under specific covariance and mean conditions.

A critical point exists where infinite divisibility transitions from true to false.

The paper characterizes the boundary of infinite divisibility for shifted Gaussian squares.

Abstract

Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq 0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ to be infinitely divisible for all $\alpha\in R^{1}$ is that \[ \Ga_{i,i}\geq \frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper we show that when this does not hold there exists an $0<\alpha_{0}<\ff $ such that $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ is infinitely divisible for all $|\alpha|\leq \alpha_{0}$ but not for any $|\al|>\al_{0}$.