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Wenming Hong

Publications and source records attributed to Wenming Hong.

At least 19 recordsLinked to original sources
arXiv2025

The Genealogy of Conditioned Critical Galton--Watson Trees

Let $T$ be the random family tree associated with the critical Galton--Watson process $\{Z_{n}\}_{n\geq0}$. Geiger (1999) provided an explicit representation of the law of $T$ conditioned on $\{Z_{n}>0\}$ by inductively constructing the conditioned tree $\tilde{T}_n$ along the line of descent of the left--most particle. Intuitively, $\tilde{T}_n$ is composed of a ``spine'', with a conditioned GW tree attached to each of its ``left'' siblings and an ordinary GW tree attached to each of its ``right'' siblings. The structure of the ``right'' subtree is partially discussed in Geiger (1999), whereas the information about the ``left'' subtree remains largely unexplored. In this paper we investigate the genealogy of the ``left'' and ``right'' subtrees of $\tilde{T}_n$, and some intrinsic properties will be specified, including the branching mechanism, the limiting behavior of the coalescent times, the limiting number of siblings of the spine, and the scaling limit distribution of the particles. We prove that the sequence of coalescent times of the ``right'' subtree converge in distribution to a sequence of ``nested uniform random variables'' by scaling, interestingly, so do the sequence of coalescent times for the ``left'' subtree after a functional transformation. We also establish the asymptotic independence of the two sequences of coalescent times, which in turn implies the asymptotic independence of the ``left'' and ``right'' subtrees. Finally, as an application, we present a probabilistic proof of the conditional limit theorem for the critical GW process established by Spitzer (unpublished) and Lamperti and Ney (1968), that is, for any fixed $0<t<1$, $Z_{[nt]}/n$ converges in distribution to the sum of two independent exponential random variables with different parameters, which exactly come from the ``left'' and ``right'' subtrees respectively.

math.PR↗
arXiv2025

On the maximal displacement of critical branching random walk in random environment

In this article, we study the maximal displacement of critical branching random walk in random environment. Let $M_n$ be the maximal displacement of a particle in generation $n$, and $Z_n$ be the total population in generation $n$, $M$ be the rightmost point ever reached by the branching random walk. Under some reasonable conditions, we prove a conditional limit theorem, \begin{equation*} \mathcal{L}\left( \dfrac{M_n}{\sqrt{\sigma} n^{\frac{3}{4}}} |Z_n>0\right) \dcon \mathcal{L}\left(A_\Lambda\right), \end{equation*} where random variable $A_\Lambda$ is related to the standard Brownian meander. And there exist some positive constant $C_1$ and $C_2$, such that \begin{equation*} C_1\leqslant\liminf\limits_{x\rightarrow\infty}x^{\frac{2}{3}}\P(M>x) \leqslant \limsup\limits_{x\rightarrow\infty} x^{\frac{2}{3}}\P(M>x) \leqslant C_2. \end{equation*} Compared with the constant environment case (Lalley and Shao (2015)), it revaels that, the conditional limit speed for $M_n$ in random environment (i.e., $n^{\frac{3}{4}}$) is significantly greater than that of constant environment case (i.e., $n^{\frac{1}{2}}$), and so is the tail probability for the $M$ (i.e., $x^{-\frac{2}{3}}$ vs $x^{-2}$). Our method is based on the path large deviation for the reduced critical branching random walk in random environment.

math.PR↗
arXiv2024

Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier

We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{\mathcal{L}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in\mathbb{N}$), the barrier is set as $\chi_n+\varepsilon n,$ where $\{\chi_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{\mathcal{L}_n\},$ the quenched survival probability (denoted by $\varrho_{\mathcal{L}}(\varepsilon)$) of the particles system will be 0 (resp., positive) when $\varepsilon\leq 0$ (resp., $\varepsilon>0$). In the present paper, we prove that $\sqrt{\varepsilon}\log\varrho_\mathcal{L}(\varepsilon)$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\varepsilon\downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.

math.PR↗
arXiv2024

Precise Large Deviations for the Total Population of Heavy-tailed Critical Branching Processes with Immigration

We focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the critical branching process with immigration $\{X_{n}\}$, when the offspring $\xi$ is regularly varying with index $\nu+1$ and the immigration $\eta$ is regularly varying with index $\delta$ $(0\leq \nu<\delta<1)$. The precise large deviation probabilities for $S_{n}$ are specified, that is, for some appropriate sequences $\{x_{n}\}$ and $\{y_{n}\}$, uniformly for $x_{n}\leq x\leq y_{n}$, $P(S_{n}>x)\sim nx^{-\delta/(1+\nu)}L(x)$, where $L(x)$ is a slowly varying function. Different from that of the subcritical case, here the upper bound $y_n$ is needed. Essentially, this is because the tail probability of the stationary distribution is determined by the offspring or the immigration in the subcritical case. But it is determined by both when the process is critical.

math.PR↗
arXiv2024

Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration

In this article we focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the subcritical branching process with immigration $\{X_{n}\}_{n\in\mathbb{N_{+}}}$, under the condition that one of the offspring $\xi$ or immigration $\eta$ is regularly varying. The tail distribution of $S_n$ is heavily dependent on that of $\xi$ and $\eta$, and a precise large deviation probability for $S_{n}$ is specified. (i)When the tail of offspring $\xi$ is lighter than immigration $\eta$, uniformly for $x\geq x_{n}$, $P(S_{n}-ES_{n}>x)\sim c_{1}nP(\eta>x)$ with some constant $c_{1}$ and sequence $\{x_{n}\}$, where $c_{1}$ is only related to the mean of offspring; (ii) When the tail of immigration $\eta$ is not heavier than offspring $\xi$, uniformly for $x\geq x_{n}$,$P(S_{n} ES_{n}>x)\sim c_{2}nP(\xi>x)$ with some constant $c_{2}$ and sequence $\{x_{n}\}$, where $c_{2}$ is related to both the mean of offspring and the mean of immigration.

math.PR↗
arXiv2024

Quasi-stationary distributions for subcritical branching Markov chains

Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting measure of particles of generation $n$. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of $(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of $(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and don't rely on Martin boundary theory.

math.PR↗
arXiv2023

Conditional central limit theorem for critical branching random walk

Consider a critical branching random walk on $\mathbb{R}$. Let $Z^{(n)}(A)$ be the number of individuals in the $n$-th generation located in $A\in \mathcal{B}(\mathbb{R})$ and $Z_{n}:=Z^{(n)}(\mathbb{R})$ denote the population of the $n$-th generation. We prove that, under some conditions, for all $x\in \mathbb{R}$, as $n\to \infty$, $$\mathcal{L}\left(\frac{Z^{(n)}(-\infty, \sqrt{n} x]}{n} ~\bigg |~ Z_{n}>0\right) \Longrightarrow\mathcal{L}\left(Y(x)\right),$$ where $\Rightarrow$ means weak convergence and $Y(x)$ is a random variable whose distribution is specified by its moments.

math.PR↗
arXiv2023

Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in time-inhomogeneous random environment (RWRE), which related the BRWRE by the many-to-one formula. The key step is to figure out Tanaka's decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting itself as well.

math.PR↗
arXiv2023

Weak Quenched Invariance Principle for Random Walk with Random Environment in Time

Consider the invariance principle for a random walk with random environment (denoted by $\mu$) in time on $\bfR$ in a weak quenched sense. We show that a sequence of the random probability measures on $\bfR$ generated by a bounded Lipschitz functional $f$ and $\mu$ will converge in distribution to another random probability measures, which is related to $f$ and two independent Brownian motions. The upper bound of the convergence rate has been obtained. We also explain that in general, this convergence can not be strengthened to the almost surely sense.

math.PR↗
arXiv2022

Random walks in time-inhomogeneous random environment conditioned to stay positive

We consider a random walk $\{S_n\}_{n\in \mathbb{N}}$ in time-inhomogeneous random environment $\xi$. For almost each realization of $\xi$, we formulate a quenched harmonic function, based on which we can define the random walk in random environment conditioned to stay positive by the Doob's $h$-transform. Furthermore, we prove a quenched invariance principle for the conditioned random walk for almost each realization of $\xi$.

math.PR↗
arXiv2022

On the barrier problem of branching random walk in time-inhomogeneous random environment

We consider a supercritical branching random walk in time-inhomogeneous random environment with a random absorption barrier, i.e.,in each generation, only the individuals born below the barrier can survive and reproduce. Assume that the random environment is i.i.d..The barrier is set as $\chi_n+an^{\alpha},$ where $a,\alpha$ are two constants and $\{\chi_n\}$ is a certain i.i.d. random walk determined by the random environment.We show that for almost surely given environment (i.e., a sequence of point processes which is a realization of the random environment), the time-inhomogeneous branching random walk under the given environment will become extinct (resp., survive with positive probability) if $\alpha<1/3$ or $\alpha=1/3, a 1/3, a>0$ or $\alpha=1/3, a>a_c$), where $a_c$ is a positive constant determined by the random environment. The rates of extinction when $\alpha<\frac{1}{3}, a\geq0$ and $\alpha=1/3, a\in(0,a_c)$ are also obtained. These extend the main results in A\"{\i}d\'{e}kon $\&$ Jaffuel (2011) and Jaffuel (2012),to the random environment case. The influence caused by the random environment have been specified.

math.PR↗
arXiv2019

Fixed points with finite mean of the smoothing transform in random environments

At each time $n\in\mathbb{N}$, let $\bar{Y}^{(n)}=(y_{1}^{(n)},y_{2}^{(n)},\cdots)$ be a random sequence of non-negative numbers that are ultimately zero in a random environment $ξ=(ξ_{n})_{n\in\mathbb{N}}$ in time, which satisfies for each $n\in\mathbb{N}$ and a.e. $ξ,~E_ξ[\sum_{i\in\mathbb{N}_{+}}y_{i}^{(n)}(ξ)]=1.$ The existence and uniqueness of the non-negative fixed points of the associated smoothing transform in random environments is considered. These fixed points are solutions of the distributional equation for $a.e.~ξ,~Z(ξ)\overset{d}{=}\sum_{i\in\mathbb{N}_{+}}y_{i}^{(0)}(ξ)Z_{i}(Tξ),$ where when given the environment $ξ$, $Z_{i}(Tξ)~(i\in\mathbb{N}_{+})$ are $i.i.d.$ non-negative random variables, and distributed the same as $Z(ξ)$. As an application, the martingale convergence of the branching random walk in random environments is given as well. The classical results by Biggins (1977) has been extended to the random environment situation.

math.PR↗
arXiv2018

Asymptotic behaviour of heavy-tailed branching processes in random environments

Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E}\log m(ξ_{0})=\infty$. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left(\barξ,Z_{n}(\barξ)\right)$ converges almost surely to a random variable $Y(\barξ)$, where $Y\in(0,1)~η$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\left\{\frac{U(\barξ,Z_{n}(\barξ))}{c_n(\barξ)}\right\}$, where $U(\barξ)$ is a slowly varying function that may depends on $\barξ$.

math.PR↗
arXiv2018

Limit theorems for the minimal position of a branching random walk in random environment

We consider a branching system of random walk in random environment (in location) in $\mathbb{N}$. We will give the exact limit value of $\frac{M_{n}}{n}$, where $M_{n}$ denotes the minimal position of branching random walk at time $n$. A key step in the proof is to transfer our branching random walks in random environment (in location) to branching random walks in random environment (in time), by use of Bramson's "branching processes within a branching process" .

math.PR↗
arXiv2018

Reduced critical Bellman-Harris branching processes for small populations

Let $\left\{ Z(t), t\geq 0\right\} $ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. Assuming that $0 0$, we study the structure of the process $% \left\{ Z(s,t),0\leq s\leq t\right\} ,$ where $Z(s,t)$ is the number of particles in the process at moment $s$ in the initial process which either survive up to moment $t$ or have a positive offspring number at this moment.

math.PR↗
arXiv2017

Limit theorems for supercritical MBPRE with linear fractional offspring distributions

We investigate the limit behavior of supercritical multitype branching processes in random environments with linear fractional offspring distributions and show that there exists a phase transition in the behavior of local probabilites of the process affected by strongly and intermediately supercritical regimes. Some conditional limit theorems can also be obtained from the representation of generating functions.

math.PR↗
arXiv2014

Scaling limit theorems for the $κ$-transient random walk in random and non-random environment

Kesten et al.( 1975) proved the stable law for the transient RWRE (here we refer it as the $κ$-transient RWRE). After that, some similar interesting properties have also been revealed for its continuous counterpart, the diffusion proces in a Brownian environment with drift $κ$. In the present paper we will investigate the connections between these two kind of models, i.e., we will construct a sequence of the $κ$-transient RWREs and prove it convergence to the diffusion proces in a Brownian environment with drift $κ$ by proper scaling. To this end, we need a counterpart convergence for the $κ$-transient random walk in non-random environment, which is interesting itself.

math.PR↗