Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean
Abstract
Description
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 $G=((G_{1}+c_{1}α)^{2},(G_{2}+c_{2}α)^{2})$ to be infinitely divisible for all $α\in R^{1}$ 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<α_{0}<\ff $ such that $G=((G_{1}+c_{1}α)^{2},(G_{2}+c_{2}α)^{2})$ is infinitely divisible for all $|α|\leq α_{0}$ but not for any $|\al|>\al_{0}$.