Discrete Time Branching Processes in Random Environment

10Multi-type BPREs

10.1. Introduction

Multi-type branching processes in constant environments are investigated by many authors in detail. However, the model of a multi-type BPRE has been investigated much less and the answers to many basic questions that are natural for the theory of branching processes, such as classification, asymptotics of survival probability and the distribution of the population size, are not known in full generality. In this chapter we give a review of the results known for this part of the branching processes theory.

Multi-type BPREs were first considered by Athreya and Karlin [ATH 71a, ATH 71b] and subsequently investigated by Weissener [WEI 71], Kaplan [KAP 74] and Tanny [TAN 81] (see also [VAT 93] for other references).

We begin the description of the multi-type BPRE (to be abbreviated as MBPRE) by introducing some standard notation for p-dimensional vectors:

• – the unit vector e_j has a j-th component that equals 1 and all others equal zero, j = 17, …, p;
• – all-zero and all-one vectors 0 = (0, …, 0), 1 = (1, …, 1);
• – for vectors x = (x₁, …, x_p) and y = (y₁, …, y_p) set

and write x ≪ y if all the components of the vector y − x are positive;

– for a matrix set

We now introduce notation related to the p-type BPREs which mimics that for the single-type ...