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Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes

Konrad Kolesko, Matthias Meiners, Ivana Tomic

Abstract

We consider the Markov renewal equation $F(t) = f(t) + \boldsymbolμ*F(t)$ for vector-valued functions $f,F: \mathbb{R} \to \mathbb{R}^{p}$ and a $p \times p$ matrix $\boldsymbolμ$ of locally finite measures $μ^{i,j}$ on $[0,\infty)$, $i,j=1,\ldots,p$. Sgibnev [Semimultiplicative estimates for the solution of the multidimensional renewal equation. {\em Izv.\ Ross.\ Akad.\ Nauk Ser.\ Mat.}, 66(3):159--174, 2002] derived an asymptotic expansion for the solution $F$ to the above equation. We give a new, more elementary proof of Sgibnev's result, which also covers the reducible case. As a corollary, we infer an asymptotic expansion for the mean of a multi-type general branching process with finite type space counted with random characteristic. Finally, some examples are discussed that illustrate phenomena of multi-type branching.

Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes

Abstract

We consider the Markov renewal equation for vector-valued functions and a matrix of locally finite measures on , . Sgibnev [Semimultiplicative estimates for the solution of the multidimensional renewal equation. {\em Izv.\ Ross.\ Akad.\ Nauk Ser.\ Mat.}, 66(3):159--174, 2002] derived an asymptotic expansion for the solution to the above equation. We give a new, more elementary proof of Sgibnev's result, which also covers the reducible case. As a corollary, we infer an asymptotic expansion for the mean of a multi-type general branching process with finite type space counted with random characteristic. Finally, some examples are discussed that illustrate phenomena of multi-type branching.

Paper Structure

This paper contains 16 sections, 17 theorems, 131 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Assumptions and preliminaries
  4. Asymptotic expansions
  5. Discussion of the assumptions and comparison with the literature
  6. Proofs
  7. Spectral radius and Malthusian parameter
  8. Existence and uniqueness
  9. Asymptotic expansions in the non-lattice case
  10. Asymptotic expansions in the lattice case
  11. Application to general branching processes
  12. The multi-type general branching process
  13. Recursive decomposition of general branching processes
  14. Examples
  15. Auxiliary results concerning matrices
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $\boldsymbol{\mu}=(\mu^{i,j})_{i,j\in[ \, p \,]}$ be a $p \times p$ matrix of locally finite measures on $[0,\infty)$ satisfying (Aass:subcritical instant offspring) with associated Markov renewal measure $\boldsymbol U$. Then the following assertions hold.

Figures (3)

  • Figure 1: $\Lambda_{\vartheta}$ is contained in a compact rectangle in the right half-plane.
  • Figure 2: Graphs of $\mathbbm{1}_{[0,\infty)}$ (dashed) and $g_{\varepsilon}$ (solid).
  • Figure 3: A piecewise continuously differentiable path that encloses all solutions $\lambda\in\Lambda_\vartheta$.

Theorems & Definitions (38)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Lemma 2.10
  • ...and 28 more