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Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces 在双曲空间上时序演化的多变量随机场
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-28 DOI: 10.1007/s10959-024-01316-6
Anatoliy Malyarenko, Emilio Porcu
{"title":"Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces","authors":"Anatoliy Malyarenko, Emilio Porcu","doi":"10.1007/s10959-024-01316-6","DOIUrl":"https://doi.org/10.1007/s10959-024-01316-6","url":null,"abstract":"<p>Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the <i>n</i>-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006954","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
Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains 几何尔格马尔可夫链加法函数的概率和矩不等式
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-18 DOI: 10.1007/s10959-024-01315-7
Alain Durmus, Eric Moulines, Alexey Naumov, Sergey Samsonov
{"title":"Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains","authors":"Alain Durmus, Eric Moulines, Alexey Naumov, Sergey Samsonov","doi":"10.1007/s10959-024-01315-7","DOIUrl":"https://doi.org/10.1007/s10959-024-01315-7","url":null,"abstract":"<p>In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in weighted total variation norm or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"63 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139902475","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
Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term 无漂移项非线性随机热方程的不变度量
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-15 DOI: 10.1007/s10959-023-01302-4
Le Chen, Nicholas Eisenberg
{"title":"Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term","authors":"Le Chen, Nicholas Eisenberg","doi":"10.1007/s10959-023-01302-4","DOIUrl":"https://doi.org/10.1007/s10959-023-01302-4","url":null,"abstract":"<p>This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation <span>(frac{partial u}{partial t} - frac{1}{2}Delta u = b(u){dot{W}})</span>, where <i>b</i> is assumed to be a globally Lipschitz continuous function and the noise <span>({dot{W}})</span> is a centered and spatially homogeneous Gaussian noise that is white in time. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function <span>(rho )</span>, which together guarantee the existence of an invariant measure in the weighted space <span>(L^2_rho ({mathbb {R}}^d))</span>. In particular, our result covers the <i>parabolic Anderson model</i> (i.e., the case when <span>(b(u) = lambda u)</span>) starting from the Dirac delta measure.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770141","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
Non-uniqueness Phase of Percolation on Reflection Groups in $${mathbb {H}^3}$$ 反射群在 $${mathbb {H}^3}$ 中的周遍非唯一性阶段
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-15 DOI: 10.1007/s10959-024-01313-9
Jan Czajkowski
{"title":"Non-uniqueness Phase of Percolation on Reflection Groups in $${mathbb {H}^3}$$","authors":"Jan Czajkowski","doi":"10.1007/s10959-024-01313-9","DOIUrl":"https://doi.org/10.1007/s10959-024-01313-9","url":null,"abstract":"<p>We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the three-dimensional hyperbolic space <span>({mathbb {H}^3})</span> corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty non-uniqueness percolation phase, i.e. that <span>(p_c &lt; p_u)</span>. This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770285","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
Fractional Skellam Process of Order k k 阶分数斯凯拉姆过程
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-12 DOI: 10.1007/s10959-024-01314-8
{"title":"Fractional Skellam Process of Order k","authors":"","doi":"10.1007/s10959-024-01314-8","DOIUrl":"https://doi.org/10.1007/s10959-024-01314-8","url":null,"abstract":"<h3>Abstract</h3> <p>We introduce and study a fractional version of the Skellam process of order <em>k</em> by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order <em>k</em> (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its long-range dependence property. Later, we consider two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes. </p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"101 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770316","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
Waiting Time for a Small Subcollection in the Coupon Collector Problem with Universal Coupon 带通用优惠券的优惠券收集器问题中小规模子收集的等待时间
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-01-21 DOI: 10.1007/s10959-023-01312-2
Jelena Jocković, Bojana Todić
{"title":"Waiting Time for a Small Subcollection in the Coupon Collector Problem with Universal Coupon","authors":"Jelena Jocković, Bojana Todić","doi":"10.1007/s10959-023-01312-2","DOIUrl":"https://doi.org/10.1007/s10959-023-01312-2","url":null,"abstract":"<p>We consider a generalization of the classical coupon collector problem, where the set of available coupons consists of standard coupons (which can be part of the collection), and two coupons with special purposes: one that speeds up the collection process and one that slows it down. We obtain several asymptotic results related to the expectation and the variance of the waiting time until a portion of the collection is sampled, as the number of standard coupons tends to infinity.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139517463","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 Note on Transience of Generalized Multi-Dimensional Excited Random Walks 广义多维受激随机漫步的瞬态说明
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-01-11 DOI: 10.1007/s10959-023-01311-3
Rodrigo B. Alves, Giulio Iacobelli, Glauco Valle
{"title":"A Note on Transience of Generalized Multi-Dimensional Excited Random Walks","authors":"Rodrigo B. Alves, Giulio Iacobelli, Glauco Valle","doi":"10.1007/s10959-023-01311-3","DOIUrl":"https://doi.org/10.1007/s10959-023-01311-3","url":null,"abstract":"<p>We consider a variant of the generalized excited random walk (GERW) in dimension <span>(dge 2)</span> where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays more slowly than <span>(n^{-beta _0})</span> (<i>n</i> is time), where <span>(beta _0)</span> depends on the transitions of the process, the GERW is transient in the direction of the drift.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"54 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139460832","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 Convergence of the Uniform Norm and Approximation for Stochastic Processes from the Space $${textbf{F}}_psi (Omega )$$ 论来自 $${textbf{F}}_psi (Omega )$$ 空间的随机过程的统一规范和逼近的收敛性
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2023-12-20 DOI: 10.1007/s10959-023-01309-x
Iryna Rozora, Yurii Mlavets, Olga Vasylyk, Volodymyr Polishchuk
{"title":"On Convergence of the Uniform Norm and Approximation for Stochastic Processes from the Space $${textbf{F}}_psi (Omega )$$","authors":"Iryna Rozora, Yurii Mlavets, Olga Vasylyk, Volodymyr Polishchuk","doi":"10.1007/s10959-023-01309-x","DOIUrl":"https://doi.org/10.1007/s10959-023-01309-x","url":null,"abstract":"<p>In this paper, we consider random variables and stochastic processes from the space <span>({textbf{F}}_psi (Omega ))</span> and study approximation problems for such processes. The method of series decomposition of a stochastic process from <span>({textbf{F}}_psi (Omega ))</span> is used to find an approximating process called a model. The rate of convergence of the model to the process in the uniform norm is investigated. We develop an approach for estimating the cut-off level of the model under given accuracy and reliability of the simulation.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138820069","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 Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities 非均质伊托过程的时变符号及相应的最大不等式
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2023-12-19 DOI: 10.1007/s10959-023-01308-y
{"title":"The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities","authors":"","doi":"10.1007/s10959-023-01308-y","DOIUrl":"https://doi.org/10.1007/s10959-023-01308-y","url":null,"abstract":"<h3>Abstract</h3> <p>The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of <em>p</em>-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process <em>cannot</em> be utilized when considering maximal inequalities.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"215 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138741372","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 Moduli of Continuity for Operator Fractional Brownian Motion 算子分数布朗运动的连续性模量
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2023-12-15 DOI: 10.1007/s10959-023-01307-z
Wensheng Wang
{"title":"The Moduli of Continuity for Operator Fractional Brownian Motion","authors":"Wensheng Wang","doi":"10.1007/s10959-023-01307-z","DOIUrl":"https://doi.org/10.1007/s10959-023-01307-z","url":null,"abstract":"<p>The almost-sure sample path behavior of the operator fractional Brownian motion with exponent <i>D</i>, including multivariate fractional Brownian motion, is investigated. In particular, the global and the local moduli of continuity of the sample paths are established. These results show that the global and the local moduli of continuity of the sample paths are completely determined by the real parts of the eigenvalues of the exponent <i>D</i>, as well as the covariance matrix at some unit vector. These results are applicable to multivariate fractional Brownian motion.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"190 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138689353","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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