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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 48 open journal systems 


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

Jay S. Rosen, CUNY
Michael B. Marcus, CUNY


Abstract
Let G(c) be an n-dimensional Gaussian vector with mean c and let G2(c) denote the n-dimensional vector with components that are the squares of the components of G(c). G(c) is said to be `associated' if whenever G2(c) is infinitely divisible, G2(ac) is infinitely divisible for all real numbers a. Necessary and sufficient conditions exist to determine whether G(c) is associated. Associated Gaussian vectors are interesting because they are related to the local times of Markov chains with 0-potential equal to the covariance of G(0)/c.
Is it possible that G2(c) is infinitely divisible for some non-zero mean c, when the corresponding Gaussian vector G(c) is not associated? We show that for all 2 dimensional Gaussian vectors that are not associated, there exists a finite real number a0> 0 such that G2(ac) is infinitely divisible if the absolute value of a is less than or equal to a0 but not if the absolute value of a is strictly greater than a0. The number a0 is called a critical point for G(c).


Full text: PDF

Pages: 1417-1455

Published on: June 28, 2009
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Electronic Journal of Probability. ISSN: 1083-6489