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).
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