Laws of large numbers for supercritical branching Gaussian processes (original) (raw)
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On laws of large numbers in L^2 for supercritical branching Markov processes beyond λ-positivity
2017
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 λ-positive systems but also some λ-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 λ-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.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
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 λ-positive systems but also some λ-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 λ-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.
On a class of Time-fractional Continuous-state Branching Processes
arXiv: Probability, 2017
We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton--Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model.
Harmonic moments and large deviation rates for supercritical branching processes
Annals of Applied Probability, 2003
Let {Z n , n ≥ 1} be a single type supercritical Galton-Watson process with mean EZ 1 ≡ m, initiated by a single ancestor. This paper studies the large deviation behavior of the sequence {R n ≡ Z n+1 Z n : n ≥ 1} and establishes a "phase transition" in rates depending on whether r, the maximal number of moments possessed by the offspring distribution, is less than, equal to or greater than the Schröder constant α. This is done via a careful analysis of the harmonic moments of Z n .
Local limit theory and large deviations for supercritical Branching processes
Annals of Applied Probability, 2004
In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to study conditional large deviations of {Y_{Z_n}:n\ge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on Z_n\ge v_n.
Stochastic Processes and their Applications, 1999
Let (Z(t): t¿0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)=EZ(t)(t → ∞) or that of Yn (n → ∞). We study the following problems: (a) the absolute continuity of the distribution of Z and the regularity of the density function; (b) the decay rate (polynomial or exponential) of the left tail probability P(Z6x) as x → 0, and that of the characteristic function Ee itZ and its derivative as |t| → ∞; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z ¿ x) as x → ∞, the analyticity of the characteristic function (t) = Ee itZ and its growth rate as an entire characteristic function. The results are established for non-trivial solutions of an associated functional equation, and are therefore also applicable for other limit variables arising in age-dependent branching processes and in homogeneous branching random walks.
Fractional Processes: From Poisson to Branching One
International Journal of Bifurcation and Chaos, 2008
Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Lèvy stable densities are discussed and used for the construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z = N/〈N〉 are found for both processes.
The extremal process of branching Brownian motion
Probability Theory and Related Fields, 2013
We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform "unusually large displacements", and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov-Petrovsky-Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process, which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses.
Extended Sibuya Distribution in Subcritical Markov Branching Processes
Proceedings of the Bulgarian Academy of Sciences
The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t > 0 follows the shifted extended Sibuya distribution, with parameters depending on the time t > 0. The conditional limit probability is the logarithmic series distribution supported by the positive integers.