Pub Date : 2023-08-01DOI: 10.1137/s0040585x97t991386
M. V. Platonova, K. S. Ryadovkin
We consider a continuous-time supercritical symmetric branching random walk on a multidimensional graph with periodic particle generation sources. A logarithmic asymptotic formula is obtained for the moments of population sizes of particles at each vertex of the graph as ${ttoinfty}$.
{"title":"Moment Asymptotics of Population Size of Particles at Vertices for a Supercritical Branching Random Walk on a Periodic Graph","authors":"M. V. Platonova, K. S. Ryadovkin","doi":"10.1137/s0040585x97t991386","DOIUrl":"https://doi.org/10.1137/s0040585x97t991386","url":null,"abstract":"We consider a continuous-time supercritical symmetric branching random walk on a multidimensional graph with periodic particle generation sources. A logarithmic asymptotic formula is obtained for the moments of population sizes of particles at each vertex of the graph as ${ttoinfty}$.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136021779","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}
Pub Date : 2023-08-01DOI: 10.1137/s0040585x97t991441
R. Belfadli, M. Chadad, M. Erraoui
In the same context as in the seminal paper [P. Cheridito, Bernoulli, 7 (2001), pp. 913--934], we are concerned with the semimartingale property of the sum of some Gaussian martingale and an independent fractional Brownian motion with Hurst parameter $H in (0,1)$. At the same time, we emphasize that the Markov property is lost even if the martingale owns it.
在与开创性论文相同的上下文中[P。Cheridito, Bernoulli, 7 (2001)， pp. 913—934]，我们关注具有Hurst参数$H in(0,1)$的高斯鞅和独立分数布朗运动的和的半鞅性质。同时，我们强调即使鞅拥有马尔可夫属性，马尔可夫属性也会丢失。
{"title":"On the Sum of Gaussian Martingale and an Independent Fractional Brownian Motion","authors":"R. Belfadli, M. Chadad, M. Erraoui","doi":"10.1137/s0040585x97t991441","DOIUrl":"https://doi.org/10.1137/s0040585x97t991441","url":null,"abstract":"In the same context as in the seminal paper [P. Cheridito, Bernoulli, 7 (2001), pp. 913--934], we are concerned with the semimartingale property of the sum of some Gaussian martingale and an independent fractional Brownian motion with Hurst parameter $H in (0,1)$. At the same time, we emphasize that the Markov property is lost even if the martingale owns it.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136021781","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}
Pub Date : 2023-05-01DOI: 10.1137/s0040585x97t991301
T. Lu, C. Ma, F. Wang
This paper provides series expansions for fractional Brownian motions on the unit ball and the unit sphere by means of ultraspherical polynomials and spherical harmonics. It establishes the property of strong local nondeterminism of isotropic Gaussian random fields on the unit sphere and that of fractional and bifractional Brownian motions on the unit ball and the unit sphere.
{"title":"Series Expansions of Fractional Brownian Motions and Strong Local Nondeterminism of Bifractional Brownian Motions on Balls and Spheres","authors":"T. Lu, C. Ma, F. Wang","doi":"10.1137/s0040585x97t991301","DOIUrl":"https://doi.org/10.1137/s0040585x97t991301","url":null,"abstract":"This paper provides series expansions for fractional Brownian motions on the unit ball and the unit sphere by means of ultraspherical polynomials and spherical harmonics. It establishes the property of strong local nondeterminism of isotropic Gaussian random fields on the unit sphere and that of fractional and bifractional Brownian motions on the unit ball and the unit sphere.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135658638","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}
Pub Date : 2023-05-01DOI: 10.1137/s0040585x97t991295
M. Dörpinghaus, I. Neri, É. Roldán, F. Jülicher
An interesting question is whether an information theoretic interpretation can be given of optimal algorithms in sequential hypothesis testing. We prove that for the binary sequential probability ratio test of a continuous observation process, the mutual information between the observation process up to the decision time and the actual hypothesis conditioned on the decision variable is equal to zero. This result can be interpreted as an optimal usage of the information on the hypothesis available in the observations by the sequential probability ratio test. As a consequence, the mutual information between the random decision time of the sequential probability ratio test and the actual hypothesis conditioned on the decision variable is also equal to zero.
{"title":"Optimal Information Usage in Binary Sequential Hypothesis Testing","authors":"M. Dörpinghaus, I. Neri, É. Roldán, F. Jülicher","doi":"10.1137/s0040585x97t991295","DOIUrl":"https://doi.org/10.1137/s0040585x97t991295","url":null,"abstract":"An interesting question is whether an information theoretic interpretation can be given of optimal algorithms in sequential hypothesis testing. We prove that for the binary sequential probability ratio test of a continuous observation process, the mutual information between the observation process up to the decision time and the actual hypothesis conditioned on the decision variable is equal to zero. This result can be interpreted as an optimal usage of the information on the hypothesis available in the observations by the sequential probability ratio test. As a consequence, the mutual information between the random decision time of the sequential probability ratio test and the actual hypothesis conditioned on the decision variable is also equal to zero.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136273940","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}
Pub Date : 2023-05-01DOI: 10.1137/s0040585x97t991337
A. N. Shiryaev, I. V. Pavlov
This is the second installment of a two-part article presenting abstracts of talks given at the 7th International Conference on Stochastic Methods (ICSM-7), held June 2--9, 2022 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. The conference was chaired by A. N. Shiryaev. Participants included leading scientists from Russia, France, Portugal, and Tadjikistan.
这是第7届国际随机方法会议(ICSM-7)的第二部分，该会议于2022年6月2日至9日在顿国立技术大学Raduga体育健身中心的Divnomorskoe (Gelendzhik镇附近)举行。会议由A. N. Shiryaev主持。与会者包括来自俄罗斯、法国、葡萄牙和塔吉克斯坦的顶尖科学家。
{"title":"Abstracts of Talks Given at the 7th International Conference on Stochastic Methods, II","authors":"A. N. Shiryaev, I. V. Pavlov","doi":"10.1137/s0040585x97t991337","DOIUrl":"https://doi.org/10.1137/s0040585x97t991337","url":null,"abstract":"This is the second installment of a two-part article presenting abstracts of talks given at the 7th International Conference on Stochastic Methods (ICSM-7), held June 2--9, 2022 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. The conference was chaired by A. N. Shiryaev. Participants included leading scientists from Russia, France, Portugal, and Tadjikistan.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135658639","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}
Pub Date : 2023-05-01DOI: 10.1137/s0040585x97t991325
A. B. Piunovskiy
In this paper we revise the theory of turnpikes in discounted Markov decision processes, prove the turnpike theorem for the undiscounted model, and apply the results to the specific random walk.KeywordsturnpikeMarkov decision processdiscounted rewardaverage rewardrandom walkstochastic knapsack problem
{"title":"Turnpikes in Finite Markov Decision Processes and Random Walk","authors":"A. B. Piunovskiy","doi":"10.1137/s0040585x97t991325","DOIUrl":"https://doi.org/10.1137/s0040585x97t991325","url":null,"abstract":"In this paper we revise the theory of turnpikes in discounted Markov decision processes, prove the turnpike theorem for the undiscounted model, and apply the results to the specific random walk.KeywordsturnpikeMarkov decision processdiscounted rewardaverage rewardrandom walkstochastic knapsack problem","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136273938","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}
Pub Date : 2023-05-01DOI: 10.1137/s0040585x97t991313
I. Pinelis
A necessary and sufficient condition for Pitman's asymptotic relative efficiency of the Kendall and Spearman correlation statistics for the independence test to be $1$ is given, in terms of certain smoothness and nondegeneracy properties of the model. Corresponding easy-to-use and broadly applicable sufficient conditions are obtained. These conditions hold for most well-known models of dependence.
{"title":"Asymptotic Relative Efficiency of the Kendall and Spearman Correlation Statistics","authors":"I. Pinelis","doi":"10.1137/s0040585x97t991313","DOIUrl":"https://doi.org/10.1137/s0040585x97t991313","url":null,"abstract":"A necessary and sufficient condition for Pitman's asymptotic relative efficiency of the Kendall and Spearman correlation statistics for the independence test to be $1$ is given, in terms of certain smoothness and nondegeneracy properties of the model. Corresponding easy-to-use and broadly applicable sufficient conditions are obtained. These conditions hold for most well-known models of dependence.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136273939","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}
Pub Date : 2023-02-01DOI: 10.1137/s0040585x97t991209
R. Aliyev, V. Bayramov
A renewal--reward process with dependent components and heavy-tailed interarrival times is investigated, and an asymptotic expansion as $ttoinfty$ for the expectation is derived.MSC codesrenewal processrenewal functionrenewal--reward processheavy-tailed distributionsubexponential distribution
{"title":"On Asymptotic Expansion for Mathematical Expectation of a Renewal--Reward Process with Dependent Components and Heavy-Tailed Interarrival Times","authors":"R. Aliyev, V. Bayramov","doi":"10.1137/s0040585x97t991209","DOIUrl":"https://doi.org/10.1137/s0040585x97t991209","url":null,"abstract":"A renewal--reward process with dependent components and heavy-tailed interarrival times is investigated, and an asymptotic expansion as $ttoinfty$ for the expectation is derived.MSC codesrenewal processrenewal functionrenewal--reward processheavy-tailed distributionsubexponential distribution","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136178386","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}
Pub Date : 2023-02-01DOI: 10.1137/s0040585x97t991167
R. Maller, S. Shemehsavar
We study exchangeable random partitions based on an underlying Dickman subordinator and the corresponding family of Poisson--Dirichlet distributions. The large sample distribution of the vector representing the block sizes and the number of blocks in a partition of ${1,2,dots,n}$ is shown to be, after norming and centering, a product of independent Poissons and a normal distribution. In a species or gene sampling situation, these quantities represent the abundances and the numbers of species or genes observed in a sample of size $n$ from the corresponding Poisson--Dirichlet distribution. We include a summary of known convergence results concerning the Dickman subordinator in this context.
{"title":"Generalized Poisson--Dirichlet Distributions Based on the Dickman Subordinator","authors":"R. Maller, S. Shemehsavar","doi":"10.1137/s0040585x97t991167","DOIUrl":"https://doi.org/10.1137/s0040585x97t991167","url":null,"abstract":"We study exchangeable random partitions based on an underlying Dickman subordinator and the corresponding family of Poisson--Dirichlet distributions. The large sample distribution of the vector representing the block sizes and the number of blocks in a partition of ${1,2,dots,n}$ is shown to be, after norming and centering, a product of independent Poissons and a normal distribution. In a species or gene sampling situation, these quantities represent the abundances and the numbers of species or genes observed in a sample of size $n$ from the corresponding Poisson--Dirichlet distribution. We include a summary of known convergence results concerning the Dickman subordinator in this context.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136178387","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}
Pub Date : 2023-02-01DOI: 10.1137/s0040585x97t991155
A. Klump, M. Kolb
Given a distribution on the positive extended real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function such that the first passage time of this function by a reflected Brownian motion has the given distribution. We combine the ideas of Ekström and Janson, which were developed within the scope of the one-sided inverse first-passage time problem, with the methods of De Masi et al., which were used in the context of free boundary problems, in order to give a different proof for the uniqueness for the two-sided inverse first-passage time problem by using a stochastic order relation. We provide criteria for qualitative properties of solutions of the inverse first-passage problem, which apply to the boundary corresponding to the exponential distribution.
{"title":"Uniqueness of the Inverse First-Passage Time Problem and the Shape of the Shiryaev Boundary","authors":"A. Klump, M. Kolb","doi":"10.1137/s0040585x97t991155","DOIUrl":"https://doi.org/10.1137/s0040585x97t991155","url":null,"abstract":"Given a distribution on the positive extended real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function such that the first passage time of this function by a reflected Brownian motion has the given distribution. We combine the ideas of Ekström and Janson, which were developed within the scope of the one-sided inverse first-passage time problem, with the methods of De Masi et al., which were used in the context of free boundary problems, in order to give a different proof for the uniqueness for the two-sided inverse first-passage time problem by using a stochastic order relation. We provide criteria for qualitative properties of solutions of the inverse first-passage problem, which apply to the boundary corresponding to the exponential distribution.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135962649","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}
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