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Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in R2 with non--zero mean

  
@article{EJP669,
	author = {Jay Rosen and Michael Marcus},
	title = {Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in R2 with non--zero mean},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {14},
	year = {2009},
	keywords = {infinite divisibility; Gaussian vectors; critical point.},
	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 the vector  $((G_{1}+c_{1}\alpha )^{2},(G_{2}+c_{2}\alpha )^{2})$ to be infinitely divisible for all $\alpha \in R^{1}$ is that   $$ \Gamma_{i,i}\ge \frac{c_{i}}{c_{j}}\Gamma_{i,j}>0\qquad\forall\,1\le i\ne j\le 2.\qquad(0.1)   $$ In this paper we show that when (0.1) does not hold there exists an $0<\alpha _{0} < \infty $ such that  $((G_{1}+c_{1}\alpha )^{2},(G_{2}+c_{2}\alpha )^{2})$ is infinitely divisible for all $|\alpha |\le \alpha _{0}$ but not for any $|\alpha | > \alpha _{0}$.},
	pages = {no. 48, 1417-1455},
	issn = {1083-6489},
	doi = {10.1214/EJP.v14-669},    
        url = {http://ejp.ejpecp.org/article/view/669}}