Spectral analysis of bilateral birth-death processes: some new explicit examples

TL;DR

This paper performs spectral analysis on bilateral birth-death processes, explicitly computing spectral matrices and orthogonal polynomials, and explores probabilistic properties like recurrence and invariant distributions.

Contribution

It provides new explicit examples of spectral matrices and orthogonal polynomials for bilateral birth-death processes, enhancing understanding of their probabilistic behaviors.

Findings

Explicit spectral matrices derived for several bilateral birth-death processes

Orthogonal polynomials associated with these processes are characterized

Probabilistic properties such as recurrence and invariant distributions are analyzed

Abstract

We consider the spectral analysis of several examples of bilateral birth-death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, like recurrence, the invariant distribution (if it exists) or the probability current.