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A moment approach to the law of large numbers for supercritical branching Markov processes

Christopher B. C. Dean, János Engländer, Emma Horton

TL;DR

This paper develops a moment-based approach to the law of large numbers for a broad class of supercritical branching Markov processes, exploiting high-order moment asymptotics. By analyzing the nonlinear semigroup and Perron–Frobenius spectral data, it proves that the normalized population converges in law to a limit W_x scaled by the dual eigenfunction, and that W_x is completely characterized by its moments via the explicit formula $\mathbb{E}[W_x^k]=k!\varphi(x)L_k(x)$. The method provides an alternative to spine-based proofs and delivers a clear route to determining the limit law through its moment sequence, with extensions to non-simple leading eigenvalues. The results apply to general BMPs with controlled offspring moments, connecting moment techniques to classical stochastic-process limits and moment problems.

Abstract

We offer a new proof of the classical law of large numbers for a general class of branching Markov processes based on the asymptotic behaviour of the moments developed in \cite{bmoments, gonzalez2022erratum}. Moreover, we show that the law of the limiting random variable, that is the almost sure limit of the classical additive martingale, is completely determined by its moments.

A moment approach to the law of large numbers for supercritical branching Markov processes

TL;DR

This paper develops a moment-based approach to the law of large numbers for a broad class of supercritical branching Markov processes, exploiting high-order moment asymptotics. By analyzing the nonlinear semigroup and Perron–Frobenius spectral data, it proves that the normalized population converges in law to a limit W_x scaled by the dual eigenfunction, and that W_x is completely characterized by its moments via the explicit formula . The method provides an alternative to spine-based proofs and delivers a clear route to determining the limit law through its moment sequence, with extensions to non-simple leading eigenvalues. The results apply to general BMPs with controlled offspring moments, connecting moment techniques to classical stochastic-process limits and moment problems.

Abstract

We offer a new proof of the classical law of large numbers for a general class of branching Markov processes based on the asymptotic behaviour of the moments developed in \cite{bmoments, gonzalez2022erratum}. Moreover, we show that the law of the limiting random variable, that is the almost sure limit of the classical additive martingale, is completely determined by its moments.

Paper Structure

This paper contains 7 sections, 5 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Branching Markov processes
  3. Assumptions and main result
  4. Preliminaries
  5. Moment convergence and the Stieltjes moment problem
  6. Random measures
  7. Proof of Theorem \ref{['theorem: WLLN']}

Key Result

Theorem 1

bmomentsgonzalez2022erratum Fix $k \ge 2$ and assume that H1 and H2k hold. For $\ell \le k$ and $t \ge 0$, define where, ${\color{black}L_1 (x)= 1}$ and we define iteratively for $k \ge 2$, with $[k_1, \dots, k_N]_k^2$ denoting the set of all non-negative $N$-tuples $(k_1, \dots, k_N)$ such that $\sum_{i = 1}^N k_i = k$ and at least two of the $k_i$ are strictly positive. Then, for all $\ell\leq

Theorems & Definitions (7)

  • Theorem
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 2.1: Theorem 8.6. in Gut.book
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:inversemulti']}