arXiv subjects

Xue Peng

Publications and source records attributed to Xue Peng.

5 recordsLinked to original sources

arXiv2025

Strong law of large numbers for a function of the local times of a transient random walk on groups

This paper presents the strong law of large numbers for a function of the local times of a transient random walk on groups, extending the research of Asymont and Korshunov for random walks on the integer lattice $\mathbb{Z}^d$. Under some weaker conditions, we prove that certain function of the local times converges almost surely and in $L^1$ and $L^2$. The proof is mainly based on the subadditive ergodic theorem.

math.PR↗

arXiv2024

Littlewood-Offord problems for the Curie-Weiss models

In this paper, we consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. Let \[Q_n^{+}:=\sup_{x\in\mathbb{R}}\sup_{v_1,v_2,\ldots,v_n\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1)),\] \[Q_n=\sup_{x\in\mathbb{R}}\sup_{|v_1|,|v_2|,\ldots,|v_n|\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1))\] where the random variables $(\varepsilon_i)_{1\leq i\leq n}$ are spins in Curie-Weiss models. We calculate the asymptotic properties of $Q_n^{+}$ and $Q_n$ as $n\to\infty$ and observe the phenomena of phase transitions. Meanwhile, we also get that $Q_n^{+}$ is attained when $v_1=v_2=\cdots=v_n=1$. And $Q_n$ is attained when one half of $(v_i)_{1\leq i\leq n}$ equals to $1$ and the other half equals to $-1$ when $n$ is even.This is a generalization of classical Littlewood-Offord problems from Rademacher random variables to possibly dependent random variables. In particular, it includes the case of general independent and identically distributed Bernoulli random variables.

math.PR↗

arXiv2023

The asymptotic behavior of rarely visited edges of the simple random walk

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely.

math.PR↗

arXiv2017

Distribution flows associated with positivity preserving coercive forms

For a given quasi-regular positivity preserving coercive form, we construct a family of ($σ$-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.

math.PR↗

arXiv2016

A Note on Uniform Nonintegrability of Random Variables

In a recent paper \cite{CHR16}, Chandra, Hu and Rosalsky introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. In this note, we introduce a weaker definition on uniform nonintegrability (W-UNI for short) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W-UNI, one of which is a W-UNI analogue of the celebrated de La Vallée Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised in \cite{CHR16}.

math.PR↗