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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
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-14 DOI: 10.1016/j.spl.2026.110651
Charles-Philippe Diez , Luís da Maia , Ivan Nourdin
{"title":"Mathematical research with GPT-5: A Malliavin–Stein experiment","authors":"Charles-Philippe Diez ,&nbsp;Luís da Maia ,&nbsp;Ivan Nourdin","doi":"10.1016/j.spl.2026.110651","DOIUrl":"10.1016/j.spl.2026.110651","url":null,"abstract":"<div><div>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 <em>qualitative</em> fourth-moment theorem to a <em>quantitative</em> 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.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110651"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a run-based δ-shock model with two critical levels 基于运行的两个临界水平δ冲击模型
IF 0.7 4区 数学
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2025-12-27 DOI: 10.1016/j.spl.2025.110632
Maxim Finkelstein , Hamed Lorvand , Reza Farhadian
{"title":"On a run-based δ-shock model with two critical levels","authors":"Maxim Finkelstein ,&nbsp;Hamed Lorvand ,&nbsp;Reza Farhadian","doi":"10.1016/j.spl.2025.110632","DOIUrl":"10.1016/j.spl.2025.110632","url":null,"abstract":"<div><div>In reliability engineering, the <span><math><mi>δ</mi></math></span>-shock model is used to study shock-exposed systems that are sensitive to the length of the time distance between consecutive shocks. When the system failure depends on a certain number of consecutive shocks with an inter-arrival time within a critical range, we are dealing with a run-based <span><math><mi>δ</mi></math></span>-shock model. In this paper, a new run-based <span><math><mi>δ</mi></math></span>-shock model is introduced, under which the system fails when an inter-arrival time is less than a critical threshold <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> for the first time or <span><math><mi>k</mi></math></span> consecutive inter-arrival times fall in the interval <span><math><mrow><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span>, for <span><math><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>. We study the probability behavior of the system’s stopping time as well as the survival of the system under the proposed model. As an illustrative example, we examine the survival of the system when the arrival of shocks follows a Poisson process. Furthermore, an example of applications is provided to illustrate possible application aspects.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110632"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-08 DOI: 10.1016/j.spl.2026.110640
Nguyen Thi The
{"title":"Lyapunov exponents of linear stochastic differential algebraic equations with properly stated leading terms","authors":"Nguyen Thi The","doi":"10.1016/j.spl.2026.110640","DOIUrl":"10.1016/j.spl.2026.110640","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110640"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-15 DOI: 10.1016/j.spl.2026.110655
Pritam Sarkar , Soumi Thakur Chakraborty , Ayan Pal
{"title":"When is an Exponentiated Pareto distribution infinitely divisible?","authors":"Pritam Sarkar ,&nbsp;Soumi Thakur Chakraborty ,&nbsp;Ayan Pal","doi":"10.1016/j.spl.2026.110655","DOIUrl":"10.1016/j.spl.2026.110655","url":null,"abstract":"<div><div>We investigate the conditions of infinite divisibility of the Exponentiated Pareto (EP) distribution supported on the entire positive half-line <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>, along with its discrete analogue defined on the set of non-negative integers. The EP distribution is defined via the cumulative distribution function (CDF) <span><math><msup><mrow><mrow><mo>[</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>]</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup></math></span> , where <span><math><mrow><mi>α</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>−</mo><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mi>λ</mi></mrow></msup><mo>,</mo><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, is the CDF of Pareto Type II (Lomax) distribution with tail parameter <span><math><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn><mo>.</mo></mrow></math></span> The discrete counterpart is defined as the integer part of a random variable <span><math><mi>X</mi></math></span> following the EP distribution. The main results assert that both the continuous and discrete versions of the EP distribution are infinitely divisible if <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>.</mo></mrow></math></span> A brief discussion of the Lévy process corresponding to the infinitely divisible case for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span> is provided along with a real world data illustration.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110655"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-09 DOI: 10.1016/j.spl.2026.110641
Thomas Royen
{"title":"A probability inequality for convolutions of MTP2-distribution functions","authors":"Thomas Royen","doi":"10.1016/j.spl.2026.110641","DOIUrl":"10.1016/j.spl.2026.110641","url":null,"abstract":"<div><div>A probability inequality is proved for n-fold convolutions of a smooth cumulative distribution function on <span><math><msubsup><mi>R</mi><mo>+</mo><mi>n</mi></msubsup></math></span>, which is multivariate totally positive of order 2 (MTP<sub>2</sub>). 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.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110641"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A uniform model selection test for semiparametric models 半参数模型的统一模型选择检验
IF 0.7 4区 数学
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-20 DOI: 10.1016/j.spl.2026.110658
Francesco Bravo
{"title":"A uniform model selection test for semiparametric models","authors":"Francesco Bravo","doi":"10.1016/j.spl.2026.110658","DOIUrl":"10.1016/j.spl.2026.110658","url":null,"abstract":"<div><div>This paper proposes a new simple test for model selection between two possibly misspecified competing semiparametric models. An important feature of the test is that it controls uniformly its size regardless as to whether the competing models are nested, overlapping or non nested and can be applied to overidentified models with weakly dependent observations.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110658"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New sufficient conditions for asymptotic stability in delayed stochastic differential equations 时滞随机微分方程渐近稳定的新充分条件
IF 0.7 4区 数学
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-07 DOI: 10.1016/j.spl.2026.110637
Dung T. Nguyen
{"title":"New sufficient conditions for asymptotic stability in delayed stochastic differential equations","authors":"Dung T. Nguyen","doi":"10.1016/j.spl.2026.110637","DOIUrl":"10.1016/j.spl.2026.110637","url":null,"abstract":"<div><div>This paper studies the asymptotic stability of delayed stochastic differential equations. Using the fixed point theorem and estimates for stochastic integrals, sufficient conditions for asymptotic convergence are derived. The results provide refinements of existing criteria while preserving a form that is convenient for verification. Finally, an example is presented to illustrate the obtained results.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110637"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145972704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Calculating the Gerber–Shiu function for Sparre Andersen process via collocation method 用配点法计算Sparre Andersen过程的Gerber-Shiu函数
IF 0.7 4区 数学
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2025-12-30 DOI: 10.1016/j.spl.2025.110633
Zan Yu
{"title":"Calculating the Gerber–Shiu function for Sparre Andersen process via collocation method","authors":"Zan Yu","doi":"10.1016/j.spl.2025.110633","DOIUrl":"10.1016/j.spl.2025.110633","url":null,"abstract":"<div><div>We develop a collocation framework for computing the Gerber–Shiu function in Sparre Andersen risk models. By discretizing it on collocation grids, the problem reduces to a well-conditioned linear algebraic system that is easy to implement. The method accommodates general claim-size distributions and requires only mild smoothness conditions. We derive convergence rates for the collocation solution and identify settings that yield higher-order accuracy. Numerical experiments across a variety of distributions and penalty functions confirm the approach’s accuracy, efficiency, and robustness.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110633"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exponential inequalities for sampling designs 抽样设计的指数不等式
IF 0.7 4区 数学
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-20 DOI: 10.1016/j.spl.2026.110654
Guillaume Chauvet , Mathieu Gerber
{"title":"Exponential inequalities for sampling designs","authors":"Guillaume Chauvet ,&nbsp;Mathieu Gerber","doi":"10.1016/j.spl.2026.110654","DOIUrl":"10.1016/j.spl.2026.110654","url":null,"abstract":"<div><div>In this work, we introduce an approach based on the martingale representation of a sampling design and Azuma–Hoeffding’s inequality to derive exponential inequalities for the difference between a Horvitz–Thompson estimator and its expectation. We derive a new exponential inequality for conditionally negatively associated (CNA) sampling designs, which is shown to improve over two existing inequalities that can be used in this context. We establish that Chao’s procedure, Tillé’s elimination procedure and the generalized Midzuno method are CNA sampling designs, and thus obtain an exponential inequality for these three sampling procedures. We show that our approach is useful beyond CNA sampling designs by deriving an exponential inequality for Brewer’s method.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110654"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Statistics & Probability Letters Pub Date : 2026-05-01 Epub Date: 2026-01-14 DOI: 10.1016/j.spl.2026.110653
Matthew Aldridge
{"title":"The law of thin processes: A law of large numbers for point processes","authors":"Matthew Aldridge","doi":"10.1016/j.spl.2026.110653","DOIUrl":"10.1016/j.spl.2026.110653","url":null,"abstract":"<div><div>If you take a superposition of <span><math><mi>n</mi></math></span> IID copies of a point process and thin that by a factor of <span><math><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></math></span>, then the resulting process tends to a Poisson process as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. 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.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"232 ","pages":"Article 110653"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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