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Annales de l’Institut Henri Poincaré D

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Volume 3, Issue 4, 2016, pp. 405–427
DOI: 10.4171/AIHPD/34

Published online: 2016-12-22

The arcsine law and an asymptotic behavior of orthogonal polynomials

Hayato Saigo[1] and Hiroki Sako[2]

(1) Nagahama Institute of Bio-Science and Technology, Japan
(2) Niigata University, Japan

Interacting Fock spaces connect the study of quantum probability theory, classical random variables, and orthogonal polynomials. They are pre-Hilbert spaces associated with creation, preservation, and annihilation processes. We prove that if three processes are asymptotically commutative, the arcsine law arises as the “large quantum number limits.” As a corollary, it is shown that for many probability measures, the asymptotic behavior of orthogonal polynomials is described by the arcsine function. A weaker form of asymptotic commutativity provides us with a discretized arcsine law, which is described by the Bessel functions of the first kind.

Keywords: Noncommutative probability arcsine law, interacting Fock space

Saigo Hayato, Sako Hiroki: The arcsine law and an asymptotic behavior of orthogonal polynomials. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3 (2016), 405-427. doi: 10.4171/AIHPD/34