The Arcsine law and an asymptotic behavior of orthogonal polynomials

TL;DR

This paper demonstrates that the arcsine law naturally emerges as a limit in quantum probability models involving orthogonal polynomials, revealing new connections between asymptotic commutativity and classical probability laws.

Contribution

It establishes a link between asymptotic commutativity in interacting Fock spaces and the emergence of the arcsine law in orthogonal polynomial asymptotics.

Findings

Arcsine law appears as a limit under asymptotic commutativity.

Orthogonal polynomial behavior is described by the arcsine function.

Discretized arcsine law arises under weaker asymptotic conditions.

Abstract

Interacting Fock space connects the study of quantum probability theory, classical random variables, and orthogonal polynomials. It is a pre-Hilbert space 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, asymptotic behavior of orthogonal polynomials is described by the arcsine function. A weaker form of asymptotic commutativity provides us a discretized arcsine law.