Geodesic
The arcsine law and an asymptotic behavior of orthogonal polynomials
Annales de l’Institut Henri Poincaré D, Tome 3 (2016) no. 4, pp. 405-427

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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 fi�rst kind.

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DOI : 10.4171/aihpd/34
Classification : 46-XX, 33-XX, 60-XX, 81-XX
Keywords: Noncommutative probability arcsine law, interacting Fock space 
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     title = {The arcsine law and an asymptotic behavior of orthogonal polynomials},
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Saigo, Hayato; Sako, Hiroki. The arcsine law and an asymptotic behavior of orthogonal polynomials. Annales de l’Institut Henri Poincaré D, Tome 3 (2016) no. 4, pp. 405-427. doi: 10.4171/aihpd/34

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