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

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in R2 with non--zero mean
2.	Creator	Author's name, affiliation, country	Jay S. Rosen; CUNY
2.	Creator	Author's name, affiliation, country	Michael B. Marcus; CUNY
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	infinite divisibility; Gaussian vectors; critical point.
3.	Subject	Subject classification	Primary 60E06; 60G15; Secondary 60E10.
4.	Description	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 $|\alpha | > \alpha _{0}$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Research of both authors was supported by grants from the National Science Foundation and PSCCUNY.
7.	Date	(YYYY-MM-DD)	2009-06-28
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/669
10.	Identifier	Digital Object Identifier	10.1214/EJP.v14-669
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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