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Branching Processes with Immigration in a Random Environment -- the Grincevičius--Grey setup

Peter Kevei

Abstract

We determine the tail asymptotics of the stationary distribution of a branching process with immigration in a random environment, when the immigration distribution dominates the offspring distribution. The assumptions are the same as in the Grincevičius--Grey theorem for the stochastic recurrence equation.

Branching Processes with Immigration in a Random Environment -- the Grincevičius--Grey setup

Abstract

We determine the tail asymptotics of the stationary distribution of a branching process with immigration in a random environment, when the immigration distribution dominates the offspring distribution. The assumptions are the same as in the Grincevičius--Grey theorem for the stochastic recurrence equation.
Paper Structure (2 sections, 4 theorems, 34 equations)

This paper contains 2 sections, 4 theorems, 34 equations.

Table of Contents

  1. Introduction and the main result
  2. Proof

Key Result

Theorem 1

Assume that there is a $\kappa > 0$ such that eq:grgr holds. Let $\ell$ be a slowly varying function. Then if and only if

Theorems & Definitions (8)

  • Theorem
  • Lemma 1
  • proof
  • Corollary
  • Lemma 2
  • proof
  • proof : Proof of the Theorem
  • Remark