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

Abstract

Let $G=(G_1,G_2)$ be a Gaussian vector in $R^2$ with $E{G}_1{G}_2\ne 0$. 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<α_{0}<\ff $ such that $G=((G_1+c_1\alpha {)}^2,(G_2+c_2\alpha {)}^2)$ is infinitely divisible for all $|\alpha |\le {\alpha }_0$ but not for any $|\al|>\al_{0}$.