Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in \(\mathbb R^{2}\) with non-zero mean. (English) Zbl 1191.60020

Summary: Let \(G(c)\) be an \(n\)-dimensional Gaussian vector with mean c and let \(G^{2}(c)\) 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 \(G^{2}(c)\) is infinitely divisible, \(G^{2}(ac)\) 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 \(G^{2}(c)\) 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 \(a_{0}> 0\) such that \(G^{2}(ac)\) is infinitely divisible if the absolute value of \(a\) is less than or equal to \(a_{0}\) but not if the absolute value of \(a\) is strictly greater than \(a_{0}\). The number \(a_{0}\) is called a critical point for \(G(c)\).


60E05 Probability distributions: general theory
60G15 Gaussian processes
60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
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