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Multitype branching processes in random environments with not strictly positive expectation matrices

Vilma Orgoványi, Károly Simon

Abstract

It is well known that under some conditions the almost sure survival probability of a multitype branching processes in random environment is positive if the Lyapunov exponent corresponding to the expectation matrices is positive, and zero if the Lyapunov exponent is negative. The goal of this note is to establish similar results when certain positivity conditions on the expectation matrices are not met. One application of such a result is to classify the positivity of Lebesgue measure of certain overlapping random self-similar sets in the line.

Multitype branching processes in random environments with not strictly positive expectation matrices

Abstract

It is well known that under some conditions the almost sure survival probability of a multitype branching processes in random environment is positive if the Lyapunov exponent corresponding to the expectation matrices is positive, and zero if the Lyapunov exponent is negative. The goal of this note is to establish similar results when certain positivity conditions on the expectation matrices are not met. One application of such a result is to classify the positivity of Lebesgue measure of certain overlapping random self-similar sets in the line.
Paper Structure (23 sections, 11 theorems, 112 equations, 3 figures)

This paper contains 23 sections, 11 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our result in a special case
  3. Corresponding literature
  4. Application to random self-similar sets
  5. Structure of the paper
  6. Notation
  7. Introduction to multitype branching processes
  8. Multitype branching processes in varying environments
  9. Alternative description of the process
  10. Multitype branching processes in a random environment
  11. Our Principal Assumptions
  12. Expectation matrices and survival probabilities
  13. Expectation matrices
  14. Survival probabilities
  15. Lyapunov and column-sum exponent
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that Under these conditions we have:

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: MBPRE
  • Definition 2.5
  • Example 2.6
  • Definition 3.1: Good set of matrices
  • Definition 3.2: Lyapunov exponent
  • Definition 3.3
  • ...and 30 more