## 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 $$G(c)$$.

### MSC:

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

### Keywords:

infinite divisibility; Gaussian vectors; critical point
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