Limit theorems for multivariate Bessel processes in the freezing regime
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Date
2018-11
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Abstract
Multivariate Bessel processes describe the stochastic dynamics of
interacting particle systems of Calogero-Moser-Sutherland type and are related
with β-Hermite and Laguerre ensembles. It was shown by Andraus, Katori,
and Miyashita that for fixed starting points, these processes admit interesting
limit laws when the multiplicities k tend to ∞, where in some cases the limits
are described by the zeros of classical Hermite and Laguerre polynomials. In
this paper we use SDEs to derive corresponding limit laws for starting points
of the form √k∙x for k→∞ with x in the interior of the corresponding Weyl
chambers. Our limit results are a.s. locally uniform in time. Moreover, in
some cases we present associated central limit theorems.
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Keywords
interacting particle systems, Calogero-Moser-Sutherland models, strong limiting laws, central limit theorems, zeros of Hermite polynomials, zeros of Laguerre polynomials, Hermite ensembles, Laguerre ensembles