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Jian-Sheng Xie

Publications and source records attributed to Jian-Sheng Xie.

9 recordsLinked to original sources
arXiv2024

On the First and the Second Borel-Cantelli Lemmas

Let $\{A_n\}_{n=1}^\infty$ be a sequence of events and let $\displaystyle S:=\sum_{n=1}^\infty 1_{A_n}$. We present in this note equivalent characterizations for the statements $\mathbb{P} (S<\infty)=1$ and $\mathbb{P} (S=\infty)=1$ respectively. These characterizations are of Borel-Cantelli lemma type and of Kochen-Stone lemma type respectively, which could be regarded as the most general version of the first and the second Borel-Cantelli Lemmas.

math.PR↗
arXiv2022

CLT, MDP and LDP for Range-Renewals of I.I.D.Samplings from an Infinite Discrete Distribution

Let $R_n$ be the number of distinct values of the $n$ simple samples from an infinite discrete distribution. In 1960 Bahadure proved $\displaystyle \lim_{n\to \infty} \frac{R_n}{\Enum R_n}=1$ in probability; Chen et al. proved the limit in the sense of almost sure convergence, along with other results. In this note we present results of CLT, MDP and LDP for $R_n$ under mild conditions.

math.PR↗
arXiv2017

Average Entropy of the Ranges for Simple Random Walks on Discrete Groups

Inspired by Benjamini et al (Ann. Inst. H. Poincaré Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the ranges. Some equivalent conditions for the vanishing of the average entropy of the ranges are given. Particularly, the average entropy of the ranges vanishes if and only if the random walk is recurrent or escaping to negative infinity without left jump. In order to characterize the recurrence further, we study the average entropy of the weighted digraphs formed by the random walk. We show that the random walk is recurrent if and only if the average entropy of the weighted digraphs vanishes.

math.PR↗
arXiv2014

Bounding Ornstein-Uhlenbeck Processes and Alikes

In this note we consider SDEs of the type $\mathrm{d} X_t=[F (X_t) -A X_t] \mathrm{d} t +D \mathrm{d} W_t$ under the assumptions that $A$'s eigenvalues are all of positive real parts and $F (\cdot)$ has slower-than-linear growth rate. It is proved that $\displaystyle \varlimsup_{t \to \infty} \frac{\|X_t\|}{\sqrt{\log t}} =\sqrt{2 λ_1}$ almost surely with $λ_1$ being the largest eigenvalue of the matrix $\displaystyle Σ:=\int_0^\infty e^{-s A} \cdot (D \cdot D^T) \cdot e^{-s A^T} \mathrm{d} s$; the discarded measure-zero set can be chosen independent of the initial values $X_0=x$.

math.PR↗
arXiv2014

Bounding the Solutions to Some SDEs via Ergodic Theory

In this note we consider autonomous SDEs admitting smooth invariant measures. We present a method in finding (almost everywhere) good bounds for $\sup \{\|X_t\|: t \in [0, T]\}$ for strong solutions $X_{\cdot}$ to such SDEs, which in many cases are optimal bounds. In some situation (especially in one-dimensional SDEs' cases), the discarded measure-zero set can be chosen to be a measure-zero set of the underlying Brownian motion uniform for all initial points $X_0=x$.

math.PR↗
arXiv2013

Range-Renewal Processes: SLLN, Power Law and Beyonds

Given $n$ samples of a regular discrete distribution $π$, we prove in this article first a serial of SLLNs results (of Dvoretzky and Erdös' type) which implies a typical power law when $π$ is heavy-tailed. Constructing a (random) graph from the ordered $n$ samples, we can establish other laws for the degree-distribution of the graph. The phenomena of small world is also discussed.

math.PR↗
arXiv2013

Range-Renewal Structure of I.I.D. Samplings

In this note the range-renewal structure of general independent and identically distributed samplings is discussed, which is a natural extension of the result in Chen et al (arXiv:1305.1829).

math.PR↗
arXiv2013

Range-Renewal Structure in Continued Fractions

Let $ω=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $ω\in (0,1)$; let $R_n (ω)$ (resp. $R_{n, \, k} (ω)$, $R_{n, \, k+} (ω)$) be the number of distinct partial quotients each of which appears at least once (resp. exactly $k$ times, at least $k$ times) in the sequence $a_1, \cdots, a_n$. In this paper it is proved that for Lebesgue almost all $ω\in (0,1)$ and all $k \geq 1$, $$ \displaystyle \lim_{n \to \infty} \frac{R_n (ω)}{\sqrt{n}}=\sqrt{\fracπ{\log 2}}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (ω)}{R_n (ω)}=\frac{C_{2 k}^k}{(2k-1) \cdot 4^k}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (ω)}{R_{n, \, k+} (ω)}=\frac{1}{2k}. $$ The Hausdorff dimensions of certain level sets about $R_n$ are discussed.

math.NT↗