Polynomial Ensembles and Pólya Frequency Functions

Yanik-Pascal Förster (), Mario Kieburg () and Holger Kösters ()
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Yanik-Pascal Förster: Bielefeld University
Mario Kieburg: Bielefeld University
Holger Kösters: Bielefeld University

Journal of Theoretical Probability, 2021, vol. 34, issue 4, 1917-1950

Abstract: Abstract We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian antisymmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here, we consider convolutions within the same class of Pólya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish–Chandra–Itzykson–Zuber integral, and we relate Pólya ensembles to Pólya frequency functions. For illustration, we give a number of explicit examples for our results.

Keywords: Probability measures on matrix spaces; Sums and products of independent random matrices; Polynomial ensembles; Additive convolution; Multiplicative convolution; Pólya frequency functions; Fourier transform; Hankel transform; Spherical transform; 60B20 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-01030-z

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