Statistics & Probability Letters最新文献

{"title":"On the asymptotic behavior of solutions to bilinear Caputo stochastic fractional differential equations","authors":"P.T. Huong,&nbsp;P.T. Anh","doi":"10.1016/j.spl.2024.110272","DOIUrl":"10.1016/j.spl.2024.110272","url":null,"abstract":"<div><p>In this paper, we focus on investigating the asymptotic behavior of solutions in a mean square sense to bilinear Caputo stochastic fractional differential equations (CSFDEs). The main tools in the proof include a variation of the constant formula for CSFDEs, the Jordan normal form of a matrix, the summation formula of Djrbashian type, and constructing a weighted norm in the associated Banach space.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110272"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002414/pdfft?md5=474d3621024386b9dd6b6eb18750084f&pid=1-s2.0-S0167715224002414-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142230197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment","authors":"Yingqiu Li ,&nbsp;Xin Zhang ,&nbsp;Zhan Lu ,&nbsp;Sheng Xiao","doi":"10.1016/j.spl.2024.110268","DOIUrl":"10.1016/j.spl.2024.110268","url":null,"abstract":"<div><p>Let <span><math><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110268"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002372/pdfft?md5=220bf7e97e493e929a1b6a021826f150&pid=1-s2.0-S0167715224002372-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the intransitivity of the win ratio","authors":"David Oakes","doi":"10.1016/j.spl.2024.110267","DOIUrl":"10.1016/j.spl.2024.110267","url":null,"abstract":"<div><p>The win-ratio analysis of controlled clinical trials uses pairwise comparisons between patients in the treatment and control group based on a primary outcome, say time to death, with indeterminacies resolved where possible by a secondary outcome, say time to hospitalization. The resulting preferences may not be transitive. Intransitivity occurs when potential follow-up times vary between patients and rankings from the primary events differ from those from secondary events. We characterize the structure of closed loops, derive some general properties of win-ratio preferences and provide simple numerical illustrations. Under realistic assumptions, unless all potential follow-up times are equal, intransitivities are certain to occur in sufficiently large samples, but their overall frequency is low.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110267"},"PeriodicalIF":0.9,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002360/pdfft?md5=fd0a2637e1a6a5c617f98f7212ce4ef9&pid=1-s2.0-S0167715224002360-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142230193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The quantile-based empirical likelihood for the difference of quantiles","authors":"Lichun Dai ,&nbsp;Pengfei Liu ,&nbsp;Yiming Liu ,&nbsp;Guangren Yang","doi":"10.1016/j.spl.2024.110252","DOIUrl":"10.1016/j.spl.2024.110252","url":null,"abstract":"<div><p>This paper aims to explore the inference of quantile differences using the quantile-based empirical likelihood (QEL) method. In contrast to traditional empirical likelihood-based approaches, the proposed method yields an explicit likelihood ratio, making it user-friendly in practical applications. Additionally, as an expansion, the comparison of quantile differences between two populations is initially considered as a measure of differences. The limiting distribution of the smoothed log-empirical likelihood ratio for both cases is theoretically derived. The paper also includes simulation studies and an analysis of a dataset comprising 6033 genes.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110252"},"PeriodicalIF":0.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002219/pdfft?md5=b3f775e874f61c09e571f1697b6573da&pid=1-s2.0-S0167715224002219-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142147898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on statistical distances for discrete log-concave measures","authors":"Arnaud Marsiglietti ,&nbsp;Puja Pandey","doi":"10.1016/j.spl.2024.110257","DOIUrl":"10.1016/j.spl.2024.110257","url":null,"abstract":"<div><p>In this note we explore how standard statistical distances are equivalent for discrete log-concave distributions. Distances include total variation distance, Wasserstein distance, and <span><math><mi>f</mi></math></span>-divergences.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110257"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002268/pdfft?md5=ffec21730d7f796c86b2e15d17d4c7a6&pid=1-s2.0-S0167715224002268-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142147897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Construction of weighted efficiency optimal designs for experiments with mixtures","authors":"Junpeng Li ,&nbsp;Guanghui Li ,&nbsp;Wei Leng ,&nbsp;Chongqi Zhang ,&nbsp;Hongyu Su","doi":"10.1016/j.spl.2024.110255","DOIUrl":"10.1016/j.spl.2024.110255","url":null,"abstract":"<div><p>This paper presents a weighted efficiency optimality criterion to obtain a compound optimal design that balances two different optimality objectives. An equivalence theorem and a search algorithm for a concave–concave combination of criteria for finding the weighted efficiency optimal design are given and applied to the Scheffé mixture model.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110255"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002244/pdfft?md5=e968c20ce5bde73ba02cd26a1bb2c451&pid=1-s2.0-S0167715224002244-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142128664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Information bounds for Gaussian copula parameter in stationary semiparametric Markov models","authors":"Xiaohong Chen ,&nbsp;Yanping Yi","doi":"10.1016/j.spl.2024.110254","DOIUrl":"10.1016/j.spl.2024.110254","url":null,"abstract":"<div><p>Let <span><math><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> be any univariate stationary first-order semiparametric Markov process generated from an unknown invariant marginal distribution and a bivariate Gaussian copula with unknown correlation coefficient <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. We prove that <span><math><mfenced><mrow><mn>1</mn><mo>−</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfenced></math></span> is the semiparametric efficient variance bound for estimating the correlation parameter <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> in any Gaussian copula generated first-order stationary Markov models. Surprisingly, this variance bound is strictly larger than <span><math><msup><mrow><mfenced><mrow><mn>1</mn><mo>−</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math></span> (when <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math></span>), which is the semiparametric efficient variance bound derived by Klaassen and Wellner (1997) for estimating the correlation parameter using any <span><math><mrow><mi>i</mi><mo>.</mo><mi>i</mi><mo>.</mo><mi>d</mi><mo>.</mo></mrow></math></span> data <span><math><msubsup><mrow><mrow><mo>{</mo><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> generated from a bivariate Gaussian copula with two unknown marginal distributions.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110254"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002232/pdfft?md5=b9102a8c83b499e2cb4081c8f8393ced&pid=1-s2.0-S0167715224002232-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142122113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homogenization for singularly perturbed stochastic wave equations with Hölder continuous coefficients","authors":"Li Yang","doi":"10.1016/j.spl.2024.110259","DOIUrl":"10.1016/j.spl.2024.110259","url":null,"abstract":"<div><p>This work is concerned with the homogenization problem for singularly perturbed stochastic wave equations. Under the assumption that the coefficients are only Hölder continuous, we prove the weak convergence of the original system to a limit equation with an extra Gaussian term by using the technique of Poisson equation in Hilbert space. The optimal convergence rate is also obtained.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110259"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002281/pdfft?md5=f4f389ac4a0c5f2cf95bbd1bf0550e2e&pid=1-s2.0-S0167715224002281-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142095719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Chainability of infinitely divisible measures","authors":"Shaul K. Bar-Lev ,&nbsp;Gérard Letac","doi":"10.1016/j.spl.2024.110256","DOIUrl":"10.1016/j.spl.2024.110256","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a positive measure on &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with Laplace transform &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; defined on a set whose interior &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is nonempty and let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be its cumulant transform. Then &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is infinitely divisible iff &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is a Laplace transform of some positive measure &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. If also &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is infinitely divisible, then &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is a Laplace transform of some positive measure &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and so forth, until we reach a &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is not infinitely divisible. If such a &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; does not exist, we say that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is infinitely chainable. We say that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is infinitely chainable of order &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; if it is infinitely chainable and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the smallest &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for which &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; In this note, we prove that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is infinitely chainable order &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; iff &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mr","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110256"},"PeriodicalIF":0.9,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002256/pdfft?md5=07337618e4ae45b99cc48ba49eb461e1&pid=1-s2.0-S0167715224002256-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142095538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cramér type moderate deviation for random walks conditioned to stay positive","authors":"Mingyang Sun","doi":"10.1016/j.spl.2024.110258","DOIUrl":"10.1016/j.spl.2024.110258","url":null,"abstract":"<div><p>We establish a Cramér type moderate deviation for random walks conditioned to stay positive, which gives the relative error for the central limit theorem proved by Iglehart (1974). Unlike the traditional technique of conjugate distributions, our approach is based on the strong approximation between random walks and Brownian motion in the same vein as Grama and Xiao (2021).</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110258"},"PeriodicalIF":0.9,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016771522400227X/pdfft?md5=33bdbf19368db3b1db5c1e8832523f3f&pid=1-s2.0-S016771522400227X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142128944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}