arXiv subjects

Shuxiong Zhang

Publications and source records attributed to Shuxiong Zhang.

11 recordsLinked to original sources

arXiv2026

A large-deviation principle for the empirical distribution of a regular branching random walk

Let $\{Z_n\}_{n\geq0}$ be a supercritical branching random walk with deterministic rooted $b$-ary tree, $b\geq2$, and symmetric displacements satisfying $\lim_{x\to+\infty}x^{-α}\log\Pp(X>x)=-λ$ with $α,λ>0$. Set $\overline Z_n(\cdot):=Z_n(\cdot)/Z_n(\mathbb{R}),~n\geq0.$ For a finite union $A$ of intervals, we establish the following full large-deviation principle: for every Borel set $Γ\subset[0,1]$, \begin{align*} -\inf_{q\inΓ^\circ}Q_A(q) &\leq \liminf_{n\to\infty}n^{-α/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\inΓ\right) &\leq \limsup_{n\to\infty}n^{-α/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\inΓ\right) &\leq -\inf_{q\in\overlineΓ}Q_A(q), \end{align*} where $Q_A$ is a good rate function on $[0,1]$. Meanwhile, we obtain large deviation probabilities for $\{\overline Z_n(\sqrt n\,A)\}_{n\geq1}.$ This strengthens the nonmatching upper and lower bounds obtained by Chen and He [Probab. Theory Related Fields 175 (2019) 255-307] for regular trees with Weibull displacements. Our method combines a large-deviation principle for rescaled displacement tree fields and exponential equivalence.

math.PR↗

arXiv2026

On empty balls of critical 2-dimensional branching random walks

Let $\{Z_n\}_{n\geq 0 }$ be a critical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesgue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In \cite{reves02}, Révész shows that if $d=1$, then $R_n/n$ converges in law to an exponential random variable as $n\to\infty$. Moreover, Révész (2002) conjectured that $$\lim_{n\to\infty}\frac{R_n}{\sqrt n}\overset{\text{law}}=\text{non-trival~distri.,}~d=2; \lim_{n\to\infty}{R_n}\overset{\text{law}}=\text{non-trival~distri.,}~d\geq3.$$ Later, Hu (2005) \cite{hu05} confirmed the case of $d\geq3$. This work confirms the case of $d=2$. It turns out that the limit distribution can be precisely characterized through the super-Brownian motion. Moreover, we also give complete results of empty balls of the branching random walk with infinite second moment offspring law. As a by-product, this article also improves the assumption of maximal displacements of branching random walks \cite[Theorem 1]{lalley2015}.

math.PR↗

arXiv2025

Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes

Let $\{η_i\}_{i\ge 1}$ be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of $\sum_{i=1}^nη_i$ has been extensively studied in the literature, this paper establishes new convergence regimes characterized by non-Poisson limits. Specifically, under a Markovian dependence structure, we show that $\sum_{i=1}^nη_i,$ under suitable scaling, converges almost surely or in distribution as $n\to\infty$ to a geometric or Gamma random variable. These results provide a new tool for analyzing the limit distributions of sums of Markovian dependent Bernoulli random variables. We demonstrate these results in several applications: determining the limiting distribution of the number of weak cutspheres for a $d(\ge3)$-dimensional standard Brownian motion; deriving the limit law for weak cutpoints of geometric Brownian motion; and analyzing how often the population size reaches a given threshold in certain branching processes, both with and without immigration.

math.PR↗

arXiv2025

On the maximal displacement of subcritical branching random walks with or without killing

Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$, and define $M := \sup_{n\geq 0} M_n$. In this paper we give asymptotics of tail probability of $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k) p_k<\infty$ and $\mathbb{E}[Xe^{γX}]<\infty$, where $γ>0$ is a constant such that $\mathbb{E}[e^{γX}]=\frac{1}{m}$ and $m=\sum_{k=0}^\infty kp_k\in (0,1)$. Moreover, we confirm the conjecture of Neuman and Zheng [Probab. Theory Related Fields. 167 (2017) 1137--1164] by establishing the existence of a critical value $m\mathbb{E}[X e^{γX}]$ such that \begin{align*} \lim_{n\to\infty}e^{γcn}\mathbb{P}(M_n\geq cn)= \left\{ \begin{aligned} &κ\in(0,1], &c\in\big(0,m\mathbb{E}[Xe^{γX}]\big); &0, &c\in\big(m\mathbb{E}[Xe^{γX}],\infty\big), \end{aligned} \right. \end{align*} where $κ$ represents the non-zero limit. Finally, we extend these results to the maximal displacement of branching random walks with killing. Interestingly, this limit can be characterized through both the global minimum of a random walk with positive drift and the maximal displacement of the branching random walk without killing.

math.PR↗

arXiv2024

Upper deviation probabilities for the range of a supercritical super-Brownian motion

Let $\{X_t\}_{t\geq 0 }$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $ψ$. Denote by $R_t:=\inf\{r>0:X_s(\{x\in \mathbb{R}^d:|x|\geq r\})=0,~\forall~0\leq s\leq t\}$ the radius of the minimal ball (centered at the origin) containing the range of $\{X_s\}_{s\geq 0 }$ up to time $t$. In \cite{Pinsky}, Pinsky proved that condition on non-extinction, $\lim_{t\to\infty}R_t/t=\sqrt{2β}$ in probability, where $β:=-ψ'(0)$. Afterwards, Engländer \cite{Englander04} studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, he \cite[Conjecture 8]{Englander04} conjectured that for $ρ>\sqrt {2β}$, $$ \lim_{t\to\infty}\frac{1}{t}\log\mathbb{P}(R_t\geq ρt)=-\left(\frac{ρ^2}{2}-β\right). $$ In this note, we confirmed this conjecture.

math.PR↗

arXiv2024

Upper deviation probabilities for level sets of a supercritical branching random walk

Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of $\{Z_n\}_{n\geq 0}$ and $I(x)$ be the large deviation rate function of the underlying random walk of $\{Z_n\}_{n\geq 0}$. It is known from [6] that under some mild conditions, for $x\in(0,x^*)$, $n^{-1}\log Z_n([nx,\infty))$ converges almost surely to $\log m- I(x)$ on the event of nonextinction as $n\to\infty$, where $x^*$ is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates of \[\mathbb{P}(Z_n([xn,\infty))\geq e^{an})\] as $n\to\infty$, where $x>0$ and $a>(\log m- I(x))^+$. This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion.

math.PR↗

arXiv2023

Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

Let $\{Z_n\}_{n\geq 0 }$ be a $d$-dimensional supercritical branching random walk started from the origin. Write $Z_n(S)$ for the number of particles located in a set $S\subset\mathbb{R}^d$ at time $n$. Denote by $R_n:=\inf\{ρ:Z_i(\{|x|\geq ρ\})=0,\forall~0\leq i\leq n\}$ the range of $\{Z_n\}_{n\geq 0 }$ before time $n$. In this work, we show that under some mild conditions $R_n/n$ converges in probability to some positive constant $x^*$ as $n\to\infty$. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates of $$ \mathbb{P}(R_n\leq xn)~\text{for}~x\in(0,x^*);~\mathbb{P}(R_n\geq xn) ~\text{for}~ x\in(x^*,\infty)$$ as $n\to\infty$. As a by-product, we confirm a conjecture of Engländer \cite{Englander04}.

math.PR↗

arXiv2022

On the empty balls of a critical or subcritical branching random walk

Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk, which completes the results of \cite{reves02} for the critical binary branching Wiener process.

math.PR↗

arXiv2022

On the empty balls of a critical super-Brownian motion

Let $\{X_t\}_{t\geq0}$ be a $d$-dimensional critical super-Brownian motion started from a Poisson random measure whose intensity is the Lebesgue measure. Denote by $R_t:=\sup\{u>0: X_t(\{x\in\mathbb{R}^d:|x|< u\})=0\}$ the radius of the largest empty ball centered at the origin of $X_t$. In this work, we prove that for $r>0$, $$\lim_{t\to\infty}\mathbb{P}\left(\frac{R_t}{t^{(1/d)\wedge(3-d)^+}}\geq r\right)=e^{-A_d(r)},$$ where $A_d(r)$ satisfies $\lim_{r\to\infty}\frac{A_d(r)}{r^{|d-2|+d\ind_{\{d=2\}}}}=C$ for some $C\in(0,\infty)$ depending only on $d$.

math.PR↗

arXiv2020

Lower deviation probabilities for level sets of the branching random walk

Given a branching random walk$\{Z_n\}_{n\geq0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)$ at generation $n$. It is known from \cite{Biggins1977} that under some mild conditions, $n^{-1}\log Z_n([θx^* n,\infty))$ converges a.s. to $\log m-I(θx^*)$, where $\log m-I(θx^*)$ is a positive constant. In this work, we investigate its lower deviation, in other words, the convergence rates of $$\mathbb{P}\left(Z_n([θx^* n,\infty))<e^{an}\right),$$ where $a\in[0,\log m-I(θx^*))$. Our results complete those in \cite{Mehmet}, \cite{Helower} and \cite{GWlower}.

math.PR↗

arXiv2020

On large deviation probabilities for empirical distribution of branching random walks with heavy tails

Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$, let $Z_n(A)$ be the number of particles located in interval $A$ at generation $n$. It is well known (e.g., \cite{biggins}) that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges a.s. to $ν(A)$ as $n\rightarrow\infty$, where $ν$ is the standard Gaussian measure. In this work, we investigate its large deviation probabilities under the condition that the step size or offspring law has heavy tail, i.e. the decay rate of $$\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$$ as $n\rightarrow\infty$, where $p\in(ν(A),1)$. Our results complete those in \cite{ChenHe} and \cite{Louidor}.

math.PR↗