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When is an Exponentiated Pareto distribution infinitely divisible? 什么时候幂帕累托分布是无限可除的？

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-15 DOI: 10.1016/j.spl.2026.110655

Pritam Sarkar , Soumi Thakur Chakraborty , Ayan Pal

We investigate the conditions of infinite divisibility of the Exponentiated Pareto (EP) distribution supported on the entire positive half-line

(0, \infty)

, along with its discrete analogue defined on the set of non-negative integers. The EP distribution is defined via the cumulative distribution function (CDF)

{[F (x)]}^{α}

, where

α > 0

and

F (x) = 1 - {(1 + x)}^{- λ}, x > 0

, is the CDF of Pareto Type II (Lomax) distribution with tail parameter

λ > 0 .

The discrete counterpart is defined as the integer part of a random variable

X

following the EP distribution. The main results assert that both the continuous and discrete versions of the EP distribution are infinitely divisible if

α \in (0, 1] .

A brief discussion of the Lévy process corresponding to the infinitely divisible case for

α \in (0, 1]

is provided along with a real world data illustration.

研究了整条正半线上（0,∞）支持的幂Pareto （EP）分布的无限可除性条件，以及它在非负整数集上定义的离散模拟。EP分布通过累积分布函数（CDF） [F(x)]α来定义，其中α>；0， F(x)=1−(1+x)−λ,x>；0为尾部参数为λ>；0的Pareto Type II （Lomax）分布的CDF。离散对应项定义为随机变量X遵循EP分布的整数部分。主要结果表明，如果α∈(0,1)，EP分布的连续和离散版本都是无限可分的。简要讨论了与α∈(0,1)无限可除情况相对应的lsamvy过程，并给出了一个真实世界的数据说明。

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引用次数: 0

Synchronization of coupled system driven by additive fractional Brownian motion 加性分数布朗运动驱动耦合系统的同步

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-15 DOI: 10.1016/j.spl.2026.110657

Meiling Zhao

This paper examines the synchronization of a stochastic coupled system driven by fractional Brownian motion with Hurst parameter

0 < H < \frac{1}{2}

. A key aspect of our analysis is the study of the uniform boundedness of the fractional Ornstein–Uhlenbeck process with a singular parameter, which serves as a crucial tool in determining the convergence rate of synchronization.

本文研究了Hurst参数为0<；H<；12的分数阶布朗运动驱动的随机耦合系统的同步问题。我们分析的一个关键方面是研究具有奇异参数的分数阶Ornstein-Uhlenbeck过程的一致有界性，这是确定同步收敛速度的关键工具。

引用次数: 0

Stochastic stability for linear autoregressive model with Gaussian innovations 高斯创新线性自回归模型的随机稳定性

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-15 DOI: 10.1016/j.spl.2026.110656

Ling-Di Wang , Yu Chen , Yong-Hua Mao , Yu-Hui Zhang

Autoregressive model is a basic one among those time series models, whose stochastic stability is crucial, and often a prerequisite, for statistical inference and other applications. As a specific type of time series model, the linear autoregressive model has garnered significant attention due to its simplicity and ease of generalization. In this paper, we present a comprehensive characterization of stability for the linear autoregressive model with Gaussian innovations, including recurrence and transience, the convergence rate in the ergodicity case, and R-stability in the transience case. The linear autoregressive models are fully classified according to their stability, which is entirely determined by their coefficients.

自回归模型是时间序列模型中的一种基本模型，其随机稳定性对统计推断和其他应用至关重要，往往是一个先决条件。线性自回归模型作为时间序列模型的一种特殊类型，因其简单易行、易于泛化而备受关注。本文给出了具有高斯创新的线性自回归模型的稳定性的综合表征，包括递归性和暂态性，遍历情况下的收敛速度，以及暂态情况下的r -稳定性。线性自回归模型根据其稳定性进行完全分类，而稳定性完全由其系数决定。

引用次数: 0

Mathematical research with GPT-5: A Malliavin–Stein experiment GPT-5的数学研究：一个Malliavin-Stein实验

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-14 DOI: 10.1016/j.spl.2026.110651

Charles-Philippe Diez , Luís da Maia , Ivan Nourdin

On August 20, 2025, GPT-5,was reported to have solved an open problem in convex optimization. Motivated by this episode, we conducted a controlled experiment in the Malliavin–Stein framework for central limit theorems. Our objective was to assess whether GPT-5 could go beyond known results by extending a qualitative fourth-moment theorem to a quantitative formulation with explicit convergence rates, both in the Gaussian and in the Poisson settings. To the best of our knowledge, the derivation of such quantitative rates had remained an open problem, in the sense that it had never been addressed in the existing literature. The present paper documents this experiment, presents the results obtained, and discusses their broader implications.

2025年8月20日，GPT-5被报道解决了凸优化中的一个开放问题。受此启发，我们在Malliavin-Stein框架下对中心极限定理进行了一个对照实验。我们的目标是评估GPT-5是否可以超越已知的结果，通过将定性的四矩定理扩展到具有显式收敛率的定量公式，在高斯和泊松设置中。据我们所知，这种定量比率的推导一直是一个悬而未决的问题，因为在现有文献中从来没有讨论过这个问题。本文记录了这个实验，给出了得到的结果，并讨论了它们更广泛的含义。

引用次数: 0

The law of thin processes: A law of large numbers for point processes 薄过程定律：点过程的大数定律

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-14 DOI: 10.1016/j.spl.2026.110653

Matthew Aldridge

If you take a superposition of

n

IID copies of a point process and thin that by a factor of

1 / n

, then the resulting process tends to a Poisson process as

n \to \infty

. We give a simple proof of this result that highlights its similarity to the law of large numbers and to the law of thin numbers of Harremoës et al.

如果你取一个点过程的n个IID副本的叠加，并将其细化为1/n，那么结果过程趋向于泊松过程，n→∞。我们给出了这个结果的一个简单证明，突出了它与大数定律和Harremoës等人的稀数定律的相似性。

引用次数: 0

Moment inequalities for a class of symmetric distributions 一类对称分布的矩不等式

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-14 DOI: 10.1016/j.spl.2026.110652

Weiwei Zhuang, Taizhong Hu

For a class of symmetrically distributed random vectors

(X, Y)

with finite mean, it is shown that

E | X + Y | \geq E | X - Y |

holds, along with its generalizations.

对于一类平均有限的对称分布随机向量（X，Y），证明了E|X+Y|≥E|X−Y|成立，并给出了它的推广。

引用次数: 0

Qualitative identification conditions for causal effects under bad controls in linear structural equation models 线性结构方程模型中不良控制下因果效应的定性识别条件

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-13 DOI: 10.1016/j.spl.2026.110642

Chie Taguchi , Manabu Kuroki

This paper considers a situation where the cause–effect relationships among random variables are represented by a linear structural equation model and the corresponding directed acyclic graph. When the causal effect is not quantitatively identifiable, we propose two novel qualitative identification conditions for causal effects under bad controls.

本文考虑了随机变量间的因果关系用线性结构方程模型和相应的有向无环图表示的情况。当因果效应不能定量识别时，我们提出了两个新的定性识别条件，用于不良控制下的因果效应。

引用次数: 0

A probability inequality for convolutions of MTP2-distribution functions mtp2分布函数卷积的概率不等式

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-09 DOI: 10.1016/j.spl.2026.110641

Thomas Royen

A probability inequality is proved for n-fold convolutions of a smooth cumulative distribution function on

R_{+}^{n}

, which is multivariate totally positive of order 2 (MTP₂). This inequality is sharper than an inequality of the same form as the Gaussian correlation inequality for distribution functions. An example are some multivariate chi-square distributions, derived from the diagonal of a Wishart matrix.

证明了一个光滑累积分布函数在R+n上的n次卷积的概率不等式，该函数是多元全正的2阶（MTP2）。这个不等式比分布函数的高斯相关不等式的相同形式的不等式更尖锐。一个例子是由Wishart矩阵的对角线导出的一些多元卡方分布。

引用次数: 0

Exponential stabilization via discrete-time feedback control for stochastic systems driven by G-Lévy process g - lsamvy过程驱动随机系统的离散时间反馈控制的指数镇定

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-08 DOI: 10.1016/j.spl.2026.110639

Bingjun Wang , Hongjun Gao , Mingxia Yuan

This paper investigates the exponential stabilization of stochastic systems driven by

G

-Lévy processes via discrete-time feedback control. We design a control input in the drift term based on discrete-time state observations and prove that there exists an upper bound

\bar{τ} > 0

for the observation step size such that the controlled system achieves both mean square and quasi-sure exponential stability for any step size

τ < \bar{τ}

. An illustrative example validates the effectiveness of the proposed control strategy.

本文通过离散时间反馈控制研究了由g - lsamvy过程驱动的随机系统的指数镇定问题。我们在漂移项中设计了一个基于离散时间状态观测的控制输入，并证明了观测步长存在一个上界τ τ >；0，使得被控系统在任意步长τ<；τ τ下均方稳定和准确定指数稳定。算例验证了所提控制策略的有效性。

引用次数: 0

Lyapunov exponents of linear stochastic differential algebraic equations with properly stated leading terms 具有适当前导项的线性随机微分代数方程的Lyapunov指数

IF 0.7 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters

Pub Date : 2026-01-08 DOI: 10.1016/j.spl.2026.110640

Nguyen Thi The

We investigate a class of stochastic differential algebraic equations with properly stated leading terms that are well-matched in a specific sense. We establish an existence and uniqueness theorem for this class and investigate the Lyapunov exponents in the linear case.

我们研究了一类随机微分代数方程，它们的前导项在特定意义上是良好匹配的。我们建立了该类的存在唯一性定理，并研究了线性情况下的Lyapunov指数。

引用次数: 0

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