On laws of large numbers in \(L^2\) for supercritical branching Markov processes beyond \(\lambda \)-positivity
type: Item
instance of: scholarly article
title: On laws of large numbers in \(L^2\) for supercritical branching Markov processes beyond \(\lambda \)-positivity
zbMATH Open document ID: 1456.60073
DOI: 10.1214/19-AIHP961
author: Matthieu Jonckheere
author: Santiago Saglietti
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2020-05-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1711.05674
full work available at URL: https://projecteuclid.org/euclid.aihp/1580720489
review text: The processes considered are supercritical Markov branching processes \((\xi_t)_{t\ge 0}\) with a general type space \((J,\mathcal{B}_J)\), a constant, i.e., type-independent branching rate, a constant local branching law with finite second moment and with possible absorbing states on \(\overline{J}\backslash J\). The aim is to provide conditions, covering as many different examples as possible, for \(L^2\)-convergence, as \(t\to\infty\), of \(\xi_t(B)/\mathbf{E}\,\xi_t(B')\) to \(\nu(B)D_\infty/\nu(B')\), where \(\xi_t(B)\) is the number of particles in \(B\) at time \(t\), \(\nu\) a measure on \((J,\mathcal{B}_J)\), \(D_\infty\) a nonnegative random variable and \(B\), \(B'\) are sets in some class \(\mathcal{C}\subseteq \mathcal{B}_J\), \(\nu(B')\ne 0\). Proceeding probabilistically and using spatial decomposition techniques, it is shown that whenever the distribution of the ``immortal particle'' process is regularly varying, as \(t\to\infty\), \(L^2\)-convergence holds if and only if a specific additive martingale associated with the branching process is bounded in \(L^2\). Given that boundedness, \(D_\infty\) is the \(L^2\)-limit of the martingale. An explicit formula for the asymptotic variance of the martingale is obtained, so that boundedness can be checked by direct computation. Conditions for \(\mathbf{P}(D_\infty> 0\mid\text{survival})=1\) are investigated and several examples discussed, among them classical \(\lambda\)-positive processes and, in particular, the non-\(\lambda\)-positive branching Brownian motion with drift and absorption at 0 introduced by \textit{H. Kesten} [Stochastic Processes Appl. 7, 9--47 (1978; Zbl 0383.60077)].
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 7199305
zbMATH Keywords: law of large numbers
zbMATH Keywords: branching Markov processes
zbMATH Keywords: spine decomposition
zbMATH Keywords: \(h\)-transform
zbMATH Keywords: \( \lambda \)-positivity
reviewed by: Heinrich Hering
MaRDI profile type: Publication
arXiv ID: 1711.05674
cites work: Subcritical contact process seen from the edge: convergence to quasi-equilibrium
cites work: Strong limit theorems for general supercritical branching processes with applications to branching diffusions
cites work: Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces
cites work: Fleming-Viot selects the minimal quasi-stationary distribution: the Galton-Watson case
cites work: Limit theorems for multitype continuous time Markov branching processes
cites work: Brunet-Derrida behavior of branching-selection particle systems on the line
cites work: Strong Local Survival of Branching Random Walks is Not Monotone
cites work: The critical contact process dies out
cites work: Effect of microscopic noise on front propagation
cites work: A Fleming-Viot particle representation of the Dirichlet Laplacian
cites work: Strong law of large numbers for supercritical superprocesses under second moment condition
cites work: An almost sure scaling limit theorem for Dawson-Watanabe superprocesses
cites work: Law of large numbers for branching symmetric hunt processes with measure-valued branching rates
cites work: Limit theorems for branching Markov processes
cites work: Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations
cites work: Branching diffusions, superdiffusions and random media
cites work: Law of large numbers for superdiffusions: the non-ergodic case
cites work: Spatial branching in random environments and with interaction
cites work: Strong law of large numbers for branching diffusions
cites work: Local extinction versus local exponential growth for spatial branching processes.
cites work: A scaling limit theorem for a class of superdiffusions
cites work: Law of large numbers for a class of superdiffusions
cites work: Quasi stationary distributions and Fleming-Viot processes in countable spaces
cites work: \(R\)-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata
cites work: Simulation of quasi-stationary distributions on countable spaces
cites work: Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one sided travelling-waves
cites work: Measure changes with extinction
cites work: A strong law of large numbers for branching processes: almost sure spine events
cites work: The many-to-few lemma and multiple spines
cites work: Branching Markov processes. I
cites work: Branching Markov processes. II
cites work: Branching Markov processes. III
cites work: Q2774021
cites work: Branching Brownian motion with absorption
cites work: Q5284190
cites work: Quasi-stationary distributions for Lévy processes
cites work: Strong law of large numbers for a class of superdiffusions
cites work: Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process
cites work: A strong law of large numbers for super-critical branching Brownian motion with absorption
cites work: Speed and fluctuations of \(N\)-particle branching Brownian motion with spatial selection
cites work: The \(\lambda\)-invariant measures of subcritical Bienaymé-Galton-Watson processes
cites work: Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes
cites work: Extensions of a Limit Theorem of Everett, Ulam and Harris on Multitype Branching Processes to a Branching Process with Countably Many Types
cites work: A note on speed of convergence to the quasi-stationary distribution
cites work: On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states
cites work: Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift
cites work: Q5667341
OpenAlex ID: W3004868962
description: scientific article