Hui He

Pruning Galton–Watson trees and tree-valued Markov processes

We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process

On large deviation rates for sums associated with Galton‒Watson processes

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Pruning of CRT-sub-trees

by pruning Galton-Watson trees and an analogous process

Stochastic equations of super-Lévy processes with general branching mechanism

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Discontinuous superprocesses with dependent spatial motion

by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process

Continuous-State Branching Processes in Lévy Random Environments

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Strong uniqueness for a class of singular SDEs for catalytic branching diffusions

run until its ascension time has a representation in terms of

Invariance principles for pruning processes of Galton-Watson trees

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Limit theorems for flows of branching processes

. A similar result was obtained by Aldous and Pitman (1998) in the special case of Poisson offspring distributions where they considered uniform pruning of Galton-Watson trees by adding marks on the edges of trees.

Rescaled Lotka–Volterra Models Converge to Super-Stable Processes

Abstract Given a supercritical Galton‒Watson process {Z n } and a positive sequence {ε n }, we study the limiting behaviors of ℙ(S Z n /Z n ≥ε n ) with sums S n of independent and identically distributed random variables X i and m=𝔼[Z 1]. We assume that we are in the Schröder case with 𝔼Z 1 log Z 1<∞ and X 1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z 1 is subexponentially distributed, we further obtain the convergence rate of Z n+1/Z n to m as n→∞.