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Hui He

Publications and source records attributed to Hui He.

54 records · Page 3Linked to original sources
arXiv2017

A tree-valued Markov processes associated with an admissible family of branching mechanisms

By studying an admissible family of branching mechanisms introduced in Li (2014), we obtain a pruning procedure on Lévy trees. Then we could construct a decreasing Lévy-CRT-valued process $\{{\mathcal T}_t\}$ by pruning Lévy trees and an analogous process $\{{\mathcal T}^*_t\}$ by pruning a critical Lévy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of $\{{\mathcal T}_t\}$ at the ascension time can be represented by $\{{\mathcal T}^*_t\}$. The results generalize those studied in Abraham and Delmas (2012).

math.PR↗
arXiv2017

On large deviation probabilities for empirical distribution of branching random walks: Schr{ö}der case and B{ö}ttcher case

Given a super-critical branching random walk on $\mathbb{R}$ started from the origin, let $Z\_n(\cdot)$ be the counting measure which counts the number of individuals at the $n$-th generation located in a given set. Under some mild conditions, it is known in \cite{B90} that for any interval $A\subset \mathbb{R}$, $\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}$ converges a.s. to $ν(A)$, where $ν$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\mathbb{P}\left(\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}-ν(A)>Δ\right),$$ for $Δ\in (0, 1-ν(A))$, in both Schr{ö}der case and B{ö}ttcher case.

math.PR↗
arXiv2017

Gromov-Hausdorff-Prokhorov convergence of vertex cut-trees of n-leaf Galton-Watson trees

In this paper we study the vertex cut-trees of Galton-Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov-Hausdorff-Prokhorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prokhorov convergence to Gromov-Hausdorff-Prokhorov remains open for their models conditioned to have $n$ vertices.

math.PR↗
arXiv2016

On Seneta-Heyde Scaling for a stable branching random walk

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an $α$-stable law with $1<α<2$. We prove that the derivative martingale $D_n$ converges to a non-trivial limit $D_\infty$ under some regular conditions. We also study the additive martingale $W_n$, and prove $n^\frac{1}αW_n$ converges in probability to a constant multiple of $D_\infty$.

math.PR↗
arXiv2016

Continuous-state branching processes in Levy random environments

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than $-1$. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcricital CBRE process with immigration.

math.PR↗
arXiv2015

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

Given a super-critical Galton-Watson process $\{Z_n\}$ and a positive sequence $\{ε_n\}$, we study the limiting behaviors of $P(S_{Z_n}/Z_n\geqε_n)$ and $P(S_{Z_n}/m^n\geqε_n) $ with sums $S_{n}$ of i.i.d. random variables $X_i$ and $m=E[Z_1]$. We assume that we are in Schröder case with $EZ_1\log Z_1<\infty$ and $X_1$ is in the domain of attraction of an $α$-stable law with $0<α<2$. As by-products, when $Z_1$ is sub-exponentially distributed, we further obtain the convergence rates of $ \frac{Z_{n+1}}{Z_n}$ to $m$ as $n\rightarrow\infty$.

math.PR↗
arXiv2014

Invariance principles for pruning processes of Galton-Watson trees

Pruning processes $(\mathcal{F}(θ),θ\geq 0)$ have been studied separately for Galton-Watson trees and for Lévy trees/forests. We establish here a limit theory that strongly connects the two studies. This solves an open problem by Abraham and Delmas, also formulated as a conjecture by Löhr, Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson forests $\mathcal{F}_n$, $n\geq 1$, in the domain of attraction of a Lévy forest $\mathcal{F}$, suitably scaled pruning processes $(\mathcal{F}_n(θ),θ\geq 0)$ converge in the Skorohod topology on cadlag functions with values in the space of (isometry classes of) locally compact real trees to limiting pruning processes. We separately treat pruning at branch points and pruning at edges. We apply our results to study ascension times and Kesten trees and forests.

math.PR↗
arXiv2013

A note on the scaling limits of contour functions of Galton-Watson trees

Recently, Abraham and Delmas constructed the distributions of super-critical Lévy trees truncated at a fixed height by connecting super-critical Lévy trees to (sub)critical Lévy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this work, using the existing works on the convergence of contour functions of (sub)critical trees, we prove that the contour functions of truncated super-critical Galton-Watson trees converge weakly to the distributions constructed by Abraham and Delmas.

math.PR↗
arXiv2012

Pruning of CRT-sub-trees

We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the Lévy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate $λ$, see Duquesne and Le Gall (2002). The tree-valued process, as $λ$ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters the pruning parameter $θ$ and the intensity $λ$. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as $θ$ increases) and description of the growing process (backward in time that is as $θ$ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A by-product of our result is that the super-critical Lévy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete Galton-Watson trees studied by Abraham, Delmas and He (2012).

math.PR↗
arXiv2012

Limit theorems for continuous time branching flows

We construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive half line studied in Li (2012).

math.PR↗
arXiv2012

Some limit theorems for flows of branching processes

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.

math.PR↗
arXiv2012

An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure

This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+β)$-stable branching and Lebesgue initial measure on $\bR$. We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put τ:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that $\bP\{τ=\infty\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/β} dt$ is finite or infinite.

math.PR↗
arXiv2012

Branching Particle Systems in Spectrally One-sided Levy Processes

We investigate the branching structure coded by the excursion above zero of a spectrally positive Levy process. The main idea is to identify the level of the Levy excursion as the time and count the number of jumps upcrossing the level. By regarding the size of a jump as the birth site of a particle, we construct a branching particle system in which the particles undergo nonlocal branchings and deterministic spatial motions to the left on the positive half line. A particle is removed from the system as soon as it reaches the origin. Then a measure-valued Borel right Markov process can be defined as the counting measures of the particle system. Its total mass evolves according to a Crump-Mode-Jagers branching process and its support represents the residual life times of those existing particles. A similar result for spectrally negative Levy process is established by a time reversal approach. Properties of the measure-valued processes can be studied via the excursions for the corresponding Levy processes.

math.PR↗
arXiv2011

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 $\{{\cal G}(u)\}$ by pruning Galton-Watson trees and an analogous process $\{{\cal G}^*(u)\}$ 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 $\{{\cal G}(u)\}$ run until its ascension time has a representation in terms of $\{{\cal G}^*(u)\}$. 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.

math.PR↗
arXiv2009

Fleming-Viot Processes in an Environment

We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line $\mbb{R}$ are constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson-Watanabe superprocesses, conditioned to have total mass one, are Fleming-Viot superprocesses.

math.PR↗
arXiv2009

Rescaled Lotka-Volterra Models Converge to Super Stable Processes

Recently, it has been shown that stochastic spatial Lotka-Volterra models when suitably rescaled can converge to a super Brownian motion. We show that the limit process could be a super stable process if the kernel of the underlying motion is in the domain of attraction of a stable law. The corresponding results in Brownian setting were proved by Cox and Perkins (2005, 2008). As applications of the convergence theorems, some new results on the asymptotics of the voter model started from single 1 at the origin are obtained which improve the results by Bramson and Griffeath (1980).

math.PR↗
arXiv2008

Discontinuous Superprocesses with Dependent Spatial Motion

We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting-branching particle systems where the spatial motions of the particles are not independent. The main work is to solve the martingale problem. When we turn to the uniqueness of the process, we generalize the localization method introduced by [D.W. Stroock, Diffusion processes associated with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32(1975) 209--244] to the measure-valued context. As for existence, we use particle system approximation and a perturbation method. This work generalizes the model introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab. 6(2001), no.25, 33 pp. (electronic)] where quadratic branching mechanism was considered. We also investigate some properties of the process.

math.PR↗