Stationary distribution for birth and death process with one-side bounded jumps

TL;DR

This paper analyzes a nonhomogeneous birth and death process with bounded jumps on the positive half lattice, deriving conditions for recurrence and explicitly formulating its stationary distribution using branching structures.

Contribution

It provides new sufficient conditions for recurrence and positive recurrence, and explicitly characterizes the stationary distribution for a process with bounded jumps and nonhomogeneous transition probabilities.

Findings

Established recurrence and positive recurrence conditions.

Derived explicit stationary distribution formula.

Utilized branching structure for analysis.

Abstract

In this paper, we study a birth and death process $\{N_t\}_{t\ge0}$ on positive half lattice, which at each discontinuity jumps at most a distance $R\ge 1$ to the right or exactly a distance $1$ to the left. The transitional probabilities at each site are nonhomogeneous. Firstly, sufficient conditions for the recurrence and positive recurrence are presented. Then by the branching structure within random walk with one-side bounded jumps set up in Hong and Wang (2013), the explicit form of the stationary distribution of the process $\{N_t\}_{t\ge0}$ is formulated.