Weighted moments and sharp convergence rates for supercritical multitype branching processes
Abstract
Let Zni=(Zni(1),⋯,Zni(d)), ngeqslant0, be a supercritical d-type branching process starting from a single particle of type i∈{1,⋯,d}, where d is a positive integer. The Kesten-Stigum theorem states that if the mean matrix M of the offspring distribution has spectral radius ρ>1 with the associated positive left and right eigenvectors u and v, then for all 1leqslant i leqslant d, the sequence Wni=⟨ Zni,u⟩/ρn u(i) forms a non-negative martingale, and hence converges almost surely to a random variable Wi as n→∞. In this paper, we investigate the asymptotic properties of the martingale {Wni}, focusing on weighted moments and sharp rates of convergence. We establish necessary and sufficient conditions for the finiteness of ϕ-moments of Wi, where ϕ(x)=xαℓ(x) with αgeqslant1 and ℓ is a positive function slowly varying at infinity. We also determine the sharp exponential rate of convergence of Wni to Wi in the ϕ-moment sense, identify a phase transition at the critical rate ρ1/2, and characterize the behavior at the critical value. Our proofs rely on the double martingale structure of the fundamental martingale and on convex inequalities for martingales.
Related
- On conservativity of multitype continuous-state branching processes
- Convergence of a Critical Multitype Branching Particle System with Mixed Finite and Infinite Mean Lifetimes
- Limit theorems for supercritical branching processes in Markovian environment with linear fractional offspring distributions
- Left tail asymptotics for the density of supercritical branching processes in random environment
- A multitype Markovian branching process with one-type population size dependence
BibTeX
@misc{xiao2026weighted,
title = {{Weighted moments and sharp convergence rates for supercritical multitype branching processes}},
author = {Sheng Xiao and Yingqiu Li and Xingang Liang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15273},
eprint = {2610.15273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15273},
}
RIS
TY - UNPB
TI - Weighted moments and sharp convergence rates for supercritical multitype branching processes
AU - Sheng Xiao
AU - Yingqiu Li
AU - Xingang Liang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15273
UR - https://arxiv.org/abs/2610.15273
AB - Let Z_n^i=(Z_n^i(1),⋯,Z_n^i(d)), ngeqslant0, be a supercritical d-type branching process starting from a single particle of type i∈{1,⋯,d}, where d is a positive integer. The Kesten-Stigum theorem states that if the mean matrix M of the offspring distribution has spectral radius ρ>1 with the associated positive left and right eigenvectors u and v, then for all 1leqslant i leqslant d, the sequence W_n^i=⟨ Z_n^i,u⟩/ρ^n u(i) forms a non-negative martingale, and hence converges almost surely to a random variable W^i as n→∞. In this paper, we investigate the asymptotic properties of the martingale {W_n^i}, focusing on weighted moments and sharp rates of convergence. We establish necessary and sufficient conditions for the finiteness of ϕ-moments of W^i, where ϕ(x)=x^αℓ(x) with αgeqslant1 and ℓ is a positive function slowly varying at infinity. We also determine the sharp exponential rate of convergence of W_n^i to W^i in the ϕ-moment sense, identify a phase transition at the critical rate ρ^1/2, and characterize the behavior at the critical value. Our proofs rely on the double martingale structure of the fundamental martingale and on convex inequalities for martingales.
ER -