Translator Disclaimer
February 2020 On laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivity
Matthieu Jonckheere, Santiago Saglietti
Ann. Inst. H. Poincaré Probab. Statist. 56(1): 265-295 (February 2020). DOI: 10.1214/19-AIHP961

Abstract

We give necessary and sufficient conditions for laws of large numbers to hold in $L^{2}$ for the empirical measure of a large class of branching Markov processes, including $\lambda $-positive systems but also some $\lambda $-transient ones, such as the branching Brownian motion with drift and absorption at $0$. This is a significant improvement over previous results on this matter, which had only dealt so far with $\lambda $-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly positive on the event of survival, and use this characterization to derive a simple method for simulating (quasi-)stationary distributions.

Nous obtenons des conditions nécessaires et suffisantes pour des lois des grands nombres dans $L^{2}$ concernant les mesures empiriques d’une large classe de processus de Markov branchants, comme le mouvement Brownien branchant avec dérive et absorption en $0$. Cela constitue un pas significatif pour ce genre de résultats qui étaient jusqu’à présent limités aux processus $\lambda $-positifs. Notre approche est purement probabiliste et est basée sur des décompositions en épine (spine) et des lemmes associés. De plus, nous caractérisons la stricte positivité de la limite quand le processus de branchement survit et utilisons cette caractérisation pour donner une méthode simple de simulation de distributions (quasi-)stationaires.

Citation

Download Citation

Matthieu Jonckheere. Santiago Saglietti. "On laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivity." Ann. Inst. H. Poincaré Probab. Statist. 56 (1) 265 - 295, February 2020. https://doi.org/10.1214/19-AIHP961

Information

Received: 16 November 2017; Revised: 27 January 2019; Accepted: 28 January 2019; Published: February 2020
First available in Project Euclid: 3 February 2020

zbMATH: 07199305
MathSciNet: MR4058988
Digital Object Identifier: 10.1214/19-AIHP961

Subjects:
Primary: 60F99, 60J80

Rights: Copyright © 2020 Institut Henri Poincaré

JOURNAL ARTICLE
31 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.56 • No. 1 • February 2020
Back to Top