Limit theorems on counting measures for a branching random walk with immigration in a random environment

TL;DR

This paper investigates the long-term behavior of particle counts in a branching random walk with immigration within a random environment, establishing various limit theorems including CLT, deviation principles, and free energy convergence.

Contribution

It introduces new limit theorems for counting measures in a branching random walk with immigration in a random environment, expanding understanding of their asymptotic properties.

Findings

Established a central limit theorem for the counting measures.

Proved a moderate deviation principle.

Demonstrated convergence of the free energy.

Abstract

We consider a branching random walk with immigration in a random environment, where the environment is a stationary and ergodic sequence indexed by time. We focus on the asymptotic properties of the sequence of measures $(Z_n)$ that count the number of particles of generation $n$ located in a Borel set of real line. In the present work, a series of limit theorems related to the above counting measures are established, including a central limit theorem, a moderate deviation principle and a large deviation result as well as a convergence theorem of the free energy.