# Subcritical branching processes in random environment with immigration stopped at zero

## TL;DR

This paper analyzes subcritical branching processes with immigration in a random environment, focusing on the tail distribution of their life periods, and proves exponential decay using change of measure and limit theorems.

## Contribution

It introduces a detailed analysis of the tail behavior of life periods in subcritical branching processes with immigration in random environments, with new probabilistic techniques.

## Key findings

- Tail distribution decays exponentially
- Established limit theorems for associated random walks
- Applied change of measure to analyze process behavior

## Abstract

We consider subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are the change of measure and some conditional limit theorems for random walks.