Journal of Theoretical Probability最新文献

{"title":"Precise Tail Behaviour of Some Dirichlet Series","authors":"Alexander Iksanov, Vitali Wachtel","doi":"10.1007/s10959-024-01318-4","DOIUrl":"https://doi.org/10.1007/s10959-024-01318-4","url":null,"abstract":"<p>Let <span>(eta _1)</span>, <span>(eta _2,ldots )</span> be independent copies of a random variable <span>(eta )</span> with zero mean and finite variance which is bounded from the right, that is, <span>(eta le b)</span> almost surely for some <span>(b&gt;0)</span>. Considering different types of the asymptotic behaviour of the probability <span>(mathbb {P}{eta in [b-x,b]})</span> as <span>(xrightarrow 0+)</span>, we derive precise tail asymptotics of the random Dirichlet series <span>(sum _{kge 1}k^{-alpha }eta _k)</span> for <span>(alpha in (1/2, 1])</span>.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034437","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}

{"title":"On Poisson Approximation","authors":"S. Y. Novak","doi":"10.1007/s10959-023-01310-4","DOIUrl":"https://doi.org/10.1007/s10959-023-01310-4","url":null,"abstract":"<p>The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. Among authors who contributed to the topic are Prokhorov, Kolmogorov, LeCam, Shorgin, Barbour, Hall, Deheuvels, Pfeifer, Roos, and many others. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc. From a theoretical point of view, the topic provides insights into Kolmogorov’s problem concerning the accuracy of approximation of the distribution of a sum of independent random variables by infinitely divisible laws. The task of establishing an estimate of the accuracy of Poisson approximation with a correct (the best possible) constant at the leading term remained open for decades. We present a solution to that problem in the case where the accuracy of approximation is evaluated in terms of the point metric. We generalise and sharpen the corresponding inequalities established by preceding authors. A new result is established for the intensively studied topic of compound Poisson approximation to the distribution of a sum of integer-valued r.v.s.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007263","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Some Optimal Conditions for the ASCLT.","authors":"István Berkes, Siegfried Hörmann","doi":"10.1007/s10959-023-01245-w","DOIUrl":"10.1007/s10959-023-01245-w","url":null,"abstract":"<p><p>Let <math><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></math> be independent random variables with <math><mrow><mi>E</mi><msub><mi>X</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></math> and <math><mrow><msubsup><mi>σ</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><mo>:</mo><mo>=</mo><mi>E</mi><msubsup><mi>X</mi><mrow><mi>k</mi></mrow><mn>2</mn></msubsup><mo><</mo><mi>∞</mi></mrow></math> <math><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></math>. Set <math><mrow><msub><mi>S</mi><mi>k</mi></msub><mo>=</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>k</mi></msub></mrow></math> and assume that <math><mrow><msubsup><mi>s</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><mo>:</mo><mo>=</mo><mi>E</mi><msubsup><mi>S</mi><mi>k</mi><mn>2</mn></msubsup><mo>→</mo><mi>∞</mi></mrow></math>. We prove that under the Kolmogorov condition <dispformula><math><mrow><mtable><mtr><mtd><mrow><mrow><mo>|</mo></mrow><msub><mi>X</mi><mi>n</mi></msub><mrow><mo>|</mo><mo>≤</mo></mrow><msub><mi>L</mi><mi>n</mi></msub><mo>,</mo><mspace></mspace><msub><mi>L</mi><mi>n</mi></msub><mo>=</mo><mi>o</mi><mrow><mo>(</mo><msub><mi>s</mi><mi>n</mi></msub><mo>/</mo><msup><mrow><mo>(</mo><mo>log</mo><mo>log</mo><msub><mi>s</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>we have <dispformula><math><mrow><mtable><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mo>log</mo><msubsup><mi>s</mi><mrow><mi>n</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msubsup><mi>σ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><msubsup><mi>s</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup></mfrac><mi>f</mi><mfenced><mfrac><msub><mi>S</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mfrac></mfenced><mo>→</mo><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt></mfrac><msub><mo>∫</mo><mi>R</mi></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></msup><mspace></mspace><mtext>d</mtext><mi>x</mi><mspace></mspace><mrow><mi>a</mi><mo>.</mo><mi>s</mi><mo>.</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>for any almost everywhere continuous function <math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math> satisfying <math><mrow><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><msup><mi>e</mi><mrow><mi>γ</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></math>, <math><mrow><mi>γ</mi><mo><</mo><mn>1</mn><mo>/</mo><mn>2<","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10927906/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42231838","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}