Unified representation of formulas for single birth processes

TL;DR

This paper introduces a unified framework for analyzing classical properties of single birth processes using a new explicit solution to the Poisson equation, covering criteria like recurrence, ergodicity, and extinction probability.

Contribution

It provides a novel explicit representation of the Poisson equation solution, enabling a unified approach to classical problems in single birth processes.

Findings

Unified criteria for recurrence, ergodicity, and extinction probability.

Explicit solutions facilitate analysis of classical properties.

Framework applicable to various single birth process problems.

Abstract

Based on a new explicit representation of the solution to the Poisson equation with respect to single birth processes, the unified treatment for various criteria on classical problems (including uniqueness, recurrence, ergodicity, exponential ergodicity, strong ergodicity, as well as extinction probability etc.) for the processes are presented.