The arcsine law and an asymptotic behavior of orthogonal polynomials

Abstract

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.