Pub Date : 2021-09-10DOI: 10.1137/s0040585x97t990630
E. Feinberg, A. Shiryaev
This paper describes the structure of solutions to Kolmogorov’s equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference “Chebyshev-200”, and it describes the results of their joined studies with Manasa Mandava (1984-2019).
{"title":"Kolmogorov's Equations for Jump Markov Processes and Their Applications to Control Problems","authors":"E. Feinberg, A. Shiryaev","doi":"10.1137/s0040585x97t990630","DOIUrl":"https://doi.org/10.1137/s0040585x97t990630","url":null,"abstract":"This paper describes the structure of solutions to Kolmogorov’s equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference “Chebyshev-200”, and it describes the results of their joined studies with Manasa Mandava (1984-2019).","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88426147","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 : 2021-08-05DOI: 10.1137/s0040585x97t990344
F. Aurzada, V. Betz, M. Lifshits
We investigate the behavior of a finite chain of Brownian particles interacting through a pairwise quadratic potential, with one end of the chain fixed and the other end pulled away at slow speed, ...
我们研究了一个有限的布朗粒子链通过一对二次势相互作用的行为，链的一端固定，另一端以缓慢的速度拉开。
{"title":"Breaking a Chain of Interacting Brownian Particles: A Gumbel Limit Theorem","authors":"F. Aurzada, V. Betz, M. Lifshits","doi":"10.1137/s0040585x97t990344","DOIUrl":"https://doi.org/10.1137/s0040585x97t990344","url":null,"abstract":"We investigate the behavior of a finite chain of Brownian particles interacting through a pairwise quadratic potential, with one end of the chain fixed and the other end pulled away at slow speed, ...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86895092","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 : 2021-08-05DOI: 10.1137/s0040585x97t990368
V. Zadorozhniy
We consider a linear inhomogeneous system of differential equations of special form with three random coefficients defined by characteristic functionals. Operator functions generated by the functio...
考虑一类具有特征泛函定义的三个随机系数的特殊形式线性非齐次微分方程系统。操作符函数由函数…
{"title":"The Expectation of a Solution of a Linear System of Differential Equations with Random Coefficients","authors":"V. Zadorozhniy","doi":"10.1137/s0040585x97t990368","DOIUrl":"https://doi.org/10.1137/s0040585x97t990368","url":null,"abstract":"We consider a linear inhomogeneous system of differential equations of special form with three random coefficients defined by characteristic functionals. Operator functions generated by the functio...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76132459","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 : 2021-08-05DOI: 10.1137/s0040585x97t990411
A. I. Rubinshtein, V. Sherstyukov
Extensions of S.N. Bernstein's example of three dependent random events of which any two are independent are considered. A complete description of such examples is given in the framework of the sym...
{"title":"On a Probabilistic Bernstein Model","authors":"A. I. Rubinshtein, V. Sherstyukov","doi":"10.1137/s0040585x97t990411","DOIUrl":"https://doi.org/10.1137/s0040585x97t990411","url":null,"abstract":"Extensions of S.N. Bernstein's example of three dependent random events of which any two are independent are considered. A complete description of such examples is given in the framework of the sym...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85689399","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 : 2021-08-05DOI: 10.1137/s0040585x97t990356
G. A. Bakay
Let $(xi(i),eta(i)) in {R}^{d+1}$, $i in {N}$, be independent and identically distributed random vectors, let $xi(i)in {R}^d$ be random vectors, let $eta(i)$ be improper nonnegative random v...
设$(xi(i)，eta(i)) in {R}^{d+1}$， $i in {N}$是独立的同分布随机向量，设$xi(i)in {R}^d$是随机向量，设$eta(i)$是反常的非负随机v…
{"title":"Large Deviations for a Terminating Compound Renewal Process","authors":"G. A. Bakay","doi":"10.1137/s0040585x97t990356","DOIUrl":"https://doi.org/10.1137/s0040585x97t990356","url":null,"abstract":"Let $(xi(i),eta(i)) in {R}^{d+1}$, $i in {N}$, be independent and identically distributed random vectors, let $xi(i)in {R}^d$ be random vectors, let $eta(i)$ be improper nonnegative random v...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88697989","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 : 2021-08-05DOI: 10.1137/s0040585x97t990332
A. Borovkov
We consider two large deviation principles (LDPs): the “ordinary” LDP (when the “strong” Cramer condition is met) and the “extended” LDP when only the standard Cramer condition is met and the devia...
{"title":"On Exact Large Deviation Principles for Compound Renewal Processes","authors":"A. Borovkov","doi":"10.1137/s0040585x97t990332","DOIUrl":"https://doi.org/10.1137/s0040585x97t990332","url":null,"abstract":"We consider two large deviation principles (LDPs): the “ordinary” LDP (when the “strong” Cramer condition is met) and the “extended” LDP when only the standard Cramer condition is met and the devia...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81080684","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 : 2021-05-06DOI: 10.1137/S0040585X97T990277
S. Nagaev
An upper estimate for the absolute value of the sum of the Spitzer series is obtained. This estimate depends explicitly on the distribution in terms of which the Spitzer series is defined.
得到了Spitzer级数和的绝对值的上限估计。这一估计明确地取决于斯皮策级数的定义所依据的分布。
{"title":"An Estimate for the Sum of the Spitzer Series and Its Generalization","authors":"S. Nagaev","doi":"10.1137/S0040585X97T990277","DOIUrl":"https://doi.org/10.1137/S0040585X97T990277","url":null,"abstract":"An upper estimate for the absolute value of the sum of the Spitzer series is obtained. This estimate depends explicitly on the distribution in terms of which the Spitzer series is defined.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46812612","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 : 2021-05-06DOI: 10.1137/S0040585X97T99023X
Y. Belopolskaya
A probabilistic approach to construction of the solution to the Cauchy problem for systems of nonlinear parabolic equations is developed. The systems under consideration can be subdivided into two ...
提出了一种构造非线性抛物方程组Cauchy问题解的概率方法。正在考虑的系统可以细分为两个。。。
{"title":"Systems of Nonlinear Backward and Forward Kolmogorov Equations: Generalized Solutions","authors":"Y. Belopolskaya","doi":"10.1137/S0040585X97T99023X","DOIUrl":"https://doi.org/10.1137/S0040585X97T99023X","url":null,"abstract":"A probabilistic approach to construction of the solution to the Cauchy problem for systems of nonlinear parabolic equations is developed. The systems under consideration can be subdivided into two ...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46845600","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 : 2021-05-06DOI: 10.1137/S0040585X97T990290
N. Slepov
In this paper we modify the Stein method and the auxiliary technique of distributional transformations of random variables. This enables us to estimate the convergence rate of distributions of norm...
本文对Stein方法和随机变量分布变换的辅助技术进行了改进。这使我们能够估计范数分布的收敛速度。
{"title":"Convergence Rate of Random Geometric Sum Distributions to the Laplace Law","authors":"N. Slepov","doi":"10.1137/S0040585X97T990290","DOIUrl":"https://doi.org/10.1137/S0040585X97T990290","url":null,"abstract":"In this paper we modify the Stein method and the auxiliary technique of distributional transformations of random variables. This enables us to estimate the convergence rate of distributions of norm...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48318775","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 : 2021-02-05DOI: 10.1137/S0040585X97T990204
V. Konakov, A. Falaleev
A random walk of a particle in ${R}^d$ is considered. The weak convergence of various transformations of trajectories of random flights with Poisson switching times was studied by Davydov and Konak...
{"title":"Convergence of Certain Classes of Random Flights in the Kantorovich Metric","authors":"V. Konakov, A. Falaleev","doi":"10.1137/S0040585X97T990204","DOIUrl":"https://doi.org/10.1137/S0040585X97T990204","url":null,"abstract":"A random walk of a particle in ${R}^d$ is considered. The weak convergence of various transformations of trajectories of random flights with Poisson switching times was studied by Davydov and Konak...","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49532514","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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