Stochastic Processes and their Applications最新文献

{"title":"Almost sure approximations and laws of iterated logarithm for signatures","authors":"Yuri Kifer","doi":"10.1016/j.spa.2025.104576","DOIUrl":"10.1016/j.spa.2025.104576","url":null,"abstract":"<div><div>We obtain strong invariance principles for normalized multiple iterated sums and integrals of the form <span><math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow><mrow><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mi>ν</mi><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mo>∑</mo></mrow><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo><</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>≤</mo><mi>N</mi><mi>t</mi></mrow></msub><mi>ξ</mi><mrow><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>⊗</mo><mo>⋯</mo><mo>⊗</mo><mi>ξ</mi><mrow><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>t</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow><mrow><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mi>ν</mi><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>≤</mo><mi>N</mi><mi>t</mi></mrow></msub><mi>ξ</mi><mrow><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>⊗</mo><mo>⋯</mo><mo>⊗</mo><mi>ξ</mi><mrow><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>)</mo></mrow><mi>d</mi><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><mi>d</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>ν</mi></mrow></msub></mrow></math></span>, where <span><math><msub><mrow><mrow><mo>{</mo><mi>ξ</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mo>−</mo><mi>∞</mi><mo><</mo><mi>k</mi><mo><</mo><mi>∞</mi></mrow></msub></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><mi>ξ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mo>−</mo><mi>∞</mi><mo><</mo><mi>s</mi><mo><</mo><mi>∞</mi></mrow></msub></math></span> are centered stationary vector processes with some weak dependence properties. These imply also laws of iterated logarithm and an almost sure central limit theorem for such objects. In the continuous time we work both under direct weak dependence assumptions and also within the suspension setup which is more appropriate for applications in dynamical systems. Similar results under substantially more restricted conditions were obtained in Friz and Kifer (2024) relying heavily on rough paths theory and notations while here we obtain these results in a more direct way","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104576"},"PeriodicalIF":1.1,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Berry-Esseen bounds for functionals of independent random variables","authors":"Qi-Man Shao ,&nbsp;Zhuo-Song Zhang","doi":"10.1016/j.spa.2025.104574","DOIUrl":"10.1016/j.spa.2025.104574","url":null,"abstract":"<div><div>We develop a new Berry–Esseen bound for functionals of independent random variables by introducing a simple form of Chatterjee’s perturbative approach. The main result is applied to the weighted triangle counts in inhomogeneous random graphs, random field Curie–Weiss model, set approximation with random tessellations and random sphere of influence graph models. The rate of convergence is the best possible.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"183 ","pages":"Article 104574"},"PeriodicalIF":1.1,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the expected ℓ∞-norm of high-dimensional martingales","authors":"Nicholas J.A. Harvey ,&nbsp;Christopher Liaw ,&nbsp;Victor S. Portella","doi":"10.1016/j.spa.2025.104575","DOIUrl":"10.1016/j.spa.2025.104575","url":null,"abstract":"<div><div>Motivated by a problem from theoretical machine learning, we show asymptotically optimal bounds on <span><math><mrow><mo>E</mo><mfenced><mrow><mspace></mspace><msub><mrow><mfenced><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>∞</mi></mrow></msub><mspace></mspace></mrow></mfenced><mo>/</mo><mo>E</mo><mfenced><mrow><mspace></mspace><msqrt><mrow><mi>τ</mi></mrow></msqrt><mspace></mspace></mrow></mfenced></mrow></math></span>, where <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> is a continuous stochastic process in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> being a Brownian motion for each <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></mrow></math></span> and <span><math><mi>τ</mi></math></span> being a stopping time such that <span><math><mrow><mo>E</mo><mfenced><mrow><mspace></mspace><msqrt><mrow><mi>τ</mi></mrow></msqrt><mspace></mspace></mrow></mfenced><mo>&lt;</mo><mi>∞</mi></mrow></math></span>. We further extend this result to the setting where the entries of <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> have smooth quadratic variation. Finally, we show a similar result for discrete-time processes using analogous techniques, together with a discrete version of Itô’s formula.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"183 ","pages":"Article 104575"},"PeriodicalIF":1.1,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local limit theorem for time-inhomogeneous functions of Markov processes","authors":"Leonid Koralov ,&nbsp;Shuo Yan","doi":"10.1016/j.spa.2025.104567","DOIUrl":"10.1016/j.spa.2025.104567","url":null,"abstract":"<div><div>In this paper, we consider a continuous-time Markov process and prove a local limit theorem for the integral of a time-inhomogeneous function of the process. One application is in the study of the fast-oscillating perturbations of linear dynamical systems.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104567"},"PeriodicalIF":1.1,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gaussian fluctuations for the wave equation under rough random perturbations","authors":"Raluca M. Balan ,&nbsp;Jingyu Huang ,&nbsp;Xiong Wang ,&nbsp;Panqiu Xia ,&nbsp;Wangjun Yuan","doi":"10.1016/j.spa.2025.104569","DOIUrl":"10.1016/j.spa.2025.104569","url":null,"abstract":"<div><div>In this article, we consider the stochastic wave equation in spatial dimension <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span>, with linear term <span><math><mrow><mi>σ</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>u</mi></mrow></math></span> multiplying the noise. This equation is driven by a Gaussian noise which is white in time and fractional in space with Hurst index <span><math><mrow><mi>H</mi><mo>∈</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mrow></math></span>. First, we prove that the solution is strictly stationary and ergodic in the spatial variable. Then, we show that with proper normalization and centering, the spatial average of the solution converges to the standard normal distribution, and we estimate the rate of this convergence in the total variation distance. We also prove the corresponding functional convergence result.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104569"},"PeriodicalIF":1.1,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The compact support property of rough super Brownian motion on R2","authors":"Ruhong Jin ,&nbsp;Nicolas Perkowski","doi":"10.1016/j.spa.2025.104568","DOIUrl":"10.1016/j.spa.2025.104568","url":null,"abstract":"<div><div>We discuss the compact support property of the rough super-Brownian motion constructed in Perkowski and Rosati (2021) as a scaling limit of a branching random walk in static random environment. The semi-linear equation corresponding to this measure-valued process is the continuous parabolic Anderson model, a singular SPDE in need of renormalization, which prevents the use of classical PDE arguments as in Englander (2006). But with the help of an interior estimation method developed in Moinat (2020), we are able to show that the compact support property also holds for rough super-Brownian motion.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104568"},"PeriodicalIF":1.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong approximations in the almost sure central limit theorem and limit behavior of the center of mass","authors":"Zhishui Hu ,&nbsp;Wei Wang ,&nbsp;Liang Dong","doi":"10.1016/j.spa.2025.104570","DOIUrl":"10.1016/j.spa.2025.104570","url":null,"abstract":"<div><div>In this paper, we establish an almost sure central limit theorem for a general random sequence under a strong approximation condition. Additionally, we derive the law of the iterated logarithm for the center of mass corresponding to a random sequence under a different strong approximation condition. Applications to step-reinforced random walks are also discussed.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104570"},"PeriodicalIF":1.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact asymptotics of ruin probabilities with linear Hawkes arrivals","authors":"Zbigniew Palmowski ,&nbsp;Simon Pojer ,&nbsp;Stefan Thonhauser","doi":"10.1016/j.spa.2025.104571","DOIUrl":"10.1016/j.spa.2025.104571","url":null,"abstract":"<div><div>In this contribution we consider a risk process whose arrivals are driven by a linear marked Hawkes process. Using an appropriate change of measure and a generalized renewal theorem, we are able to derive the exact asymptotics of the process’s ruin probability in the case of light-tailed claims. On the other hand, we can exploit the principle of one large jump to derive the analogous result in the heavy-tailed situation. Furthermore, we derive several intermediate results like the Harris recurrence of the Hawkes intensity process which are of their own interest.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104571"},"PeriodicalIF":1.1,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local weak limits for collapsed branching processes with random out-degrees","authors":"Sayan Banerjee,&nbsp;Prabhanka Deka,&nbsp;Mariana Olvera-Cravioto","doi":"10.1016/j.spa.2025.104566","DOIUrl":"10.1016/j.spa.2025.104566","url":null,"abstract":"<div><div>We obtain local weak limits in probability for Collapsed Branching Processes (CBP), which are directed random networks obtained by collapsing random-sized families of individuals in a general continuous-time branching process. The local weak limit of a given CBP, as the network grows, is shown to be a related continuous-time branching process stopped at an independent exponential time. The proof involves the construction of an explicit coupling of the in-components of vertices with the limiting object. We also show that the in-components of a finite collection of uniformly chosen vertices locally weakly converge (in probability) to i.i.d. copies of the above limit, reminiscent of propagation of chaos in interacting particle systems. We obtain as special cases novel descriptions of the local weak limits of directed preferential and uniform attachment models. We also outline some applications of our results for analyzing the limiting in-degree and PageRank distributions. In particular, upper and lower bounds on the tail of the in-degree distribution are obtained and a phase transition is detected in terms of the growth rate of the attachment function governing reproduction rates in the branching process.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104566"},"PeriodicalIF":1.1,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limit theorems for the site frequency spectrum of neutral mutations in an exponentially growing population","authors":"Einar Bjarki Gunnarsson ,&nbsp;Kevin Leder ,&nbsp;Xuanming Zhang","doi":"10.1016/j.spa.2025.104565","DOIUrl":"10.1016/j.spa.2025.104565","url":null,"abstract":"<div><div>The site frequency spectrum (SFS) is a widely used summary statistic of genomic data. Motivated by recent evidence for the role of neutral evolution in cancer, we investigate the SFS of neutral mutations in an exponentially growing population. Using branching process techniques, we establish (first-order) almost sure convergence results for the SFS of a Galton–Watson process, evaluated either at a fixed time or at the stochastic time at which the population first reaches a certain size. We finally use our results to construct consistent estimators for the extinction probability and the effective mutation rate of a birth–death process.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"182 ","pages":"Article 104565"},"PeriodicalIF":1.1,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}