Pub Date : 2024-05-02DOI: 10.1137/s0040585x97t991763
K. S. Ryadovkin
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 88-98, May 2024. We consider periodic branching random walks with periodic branching sources. It is assumed that the transition intensities of the random walk satisfy some symmetry conditions and obey a condition which ensures infinite variance of jumps. In this case, we obtain the leading term for the asymptotics of the mean population size of particles at an arbitrary point of the lattice for large time.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 1 期，第 88-98 页，2024 年 5 月。 我们考虑具有周期性分支源的周期性分支随机游走。假定随机游走的过渡强度满足某些对称条件，并服从一个确保跳跃方差无限大的条件。在这种情况下，我们得到了大时间内晶格任意点上粒子平均种群数量渐近的前导项。
{"title":"On a Periodic Branching Random Walk on $mathbf{Z}^{{d}}$ with an Infinite Variance of Jumps","authors":"K. S. Ryadovkin","doi":"10.1137/s0040585x97t991763","DOIUrl":"https://doi.org/10.1137/s0040585x97t991763","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 88-98, May 2024. <br/> We consider periodic branching random walks with periodic branching sources. It is assumed that the transition intensities of the random walk satisfy some symmetry conditions and obey a condition which ensures infinite variance of jumps. In this case, we obtain the leading term for the asymptotics of the mean population size of particles at an arbitrary point of the lattice for large time.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839311","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 : 2024-05-02DOI: 10.1137/s0040585x97t991805
M. Taniguchi, Y. Xue
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 150-160, May 2024. For Gaussian stationary processes, a time series Hellinger distance $T(f,g)$ for spectra $f$ and $g$ is derived. Evaluating $T(f_theta,f_{theta+h})$ of the form $O(h^alpha)$, we give $1/alpha$-consistent asymptotics of the maximum likelihood estimator of $theta$ for nonregular spectra. For regular spectra, we introduce the minimum Hellinger distance estimator $widehat{theta}=operatorname{arg}min_theta T(f_theta,widehat{g}_n)$, where $widehat{g}_n$ is a nonparametric spectral density estimator. We show that $widehattheta$ is asymptotically efficient and more robust than the Whittle estimator. Brief numerical studies are provided.
{"title":"Hellinger Distance Estimation for Nonregular Spectra","authors":"M. Taniguchi, Y. Xue","doi":"10.1137/s0040585x97t991805","DOIUrl":"https://doi.org/10.1137/s0040585x97t991805","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 150-160, May 2024. <br/> For Gaussian stationary processes, a time series Hellinger distance $T(f,g)$ for spectra $f$ and $g$ is derived. Evaluating $T(f_theta,f_{theta+h})$ of the form $O(h^alpha)$, we give $1/alpha$-consistent asymptotics of the maximum likelihood estimator of $theta$ for nonregular spectra. For regular spectra, we introduce the minimum Hellinger distance estimator $widehat{theta}=operatorname{arg}min_theta T(f_theta,widehat{g}_n)$, where $widehat{g}_n$ is a nonparametric spectral density estimator. We show that $widehattheta$ is asymptotically efficient and more robust than the Whittle estimator. Brief numerical studies are provided.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839537","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 : 2024-05-02DOI: 10.1137/s0040585x97t991726
A. V. Bulinski
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 25-34, May 2024. In this paper, we prove that the monotonicity property of the stability measure for the feature (factor) selection introduced by Nogueira, Sechidis, and Brown [J. Mach. Learn. Res., 18 (2018), pp. 1--54] may not hold. Another monotonicity property takes place. We also show the cases in which it is possible to compare by certain parameters the matrices describing the operation of algorithms for identifying relevant features.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 1 期，第 25-34 页，2024 年 5 月。 在本文中，我们证明了 Nogueira、Sechidis 和 Brown [J. Mach. Learn. Res., 18 (2018), pp.另一个单调性属性发生了。我们还展示了可以通过某些参数比较描述识别相关特征的算法操作矩阵的情况。
{"title":"Stability Properties of Feature Selection Measures","authors":"A. V. Bulinski","doi":"10.1137/s0040585x97t991726","DOIUrl":"https://doi.org/10.1137/s0040585x97t991726","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 25-34, May 2024. <br/> In this paper, we prove that the monotonicity property of the stability measure for the feature (factor) selection introduced by Nogueira, Sechidis, and Brown [J. Mach. Learn. Res., 18 (2018), pp. 1--54] may not hold. Another monotonicity property takes place. We also show the cases in which it is possible to compare by certain parameters the matrices describing the operation of algorithms for identifying relevant features.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839614","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 : 2024-05-02DOI: 10.1137/s0040585x97t991829
S.A. Molchanov, D.D. Sokolov, E.B. Yarovaya
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 166-167, May 2024. A remembrance of Professor Valerii Nikolaevich Tutubalin, who passed away on June 18, 2023. He held a position in the Department of Probability Theory at Moscow State University since 1965 and was known as a leading authority on probability theory and its applications.
概率论及其应用》（Theory of Probability &Its Applications），第69卷第1期，第166-167页，2024年5月。 缅怀于 2023 年 6 月 18 日逝世的瓦列里-尼古拉耶维奇-图图巴林教授。瓦列里-尼古拉耶维奇-图图巴林教授于 2023 年 6 月 18 日逝世，他自 1965 年起在莫斯科国立大学概率论系任职，是著名的概率论及其应用权威。
{"title":"In Memory of V. N. Tutubalin (10.15.1936--6.18.2023)","authors":"S.A. Molchanov, D.D. Sokolov, E.B. Yarovaya","doi":"10.1137/s0040585x97t991829","DOIUrl":"https://doi.org/10.1137/s0040585x97t991829","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 166-167, May 2024. <br/> A remembrance of Professor Valerii Nikolaevich Tutubalin, who passed away on June 18, 2023. He held a position in the Department of Probability Theory at Moscow State University since 1965 and was known as a leading authority on probability theory and its applications.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839355","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 : 2024-05-02DOI: 10.1137/s0040585x97t991751
A. V. Lyulintsev
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 71-87, May 2024. We consider a continuous-time homogeneous Markov process on the state space $mathbf{Z}_+={0,1,2,dots}$. The process is interpreted as the motion of a particle. A particle may transit only to neighboring points $mathbf{Z}_+$, i.e., for each single motion of the particle, its coordinate changes by 1. The process is equipped with a branching mechanism. Branching sources may be located at each point of $mathbf{Z}_+$. At a moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) by the same rules as the initial particle. To such a branching Markov process there corresponds a Jacobi matrix. In terms of orthogonal polynomials corresponding to this matrix, we obtain formulas for the mean number of particles at an arbitrary fixed point of $mathbf{Z}_+$ at time $t>0$. The results obtained are applied to some concrete models, an exact value for the mean number of particles in terms of special functions is given, and an asymptotic formula for this quantity for large time is found.
{"title":"Markov Branching Random Walks on $mathbf{Z}_+$: An Approach Using Orthogonal Polynomials. I","authors":"A. V. Lyulintsev","doi":"10.1137/s0040585x97t991751","DOIUrl":"https://doi.org/10.1137/s0040585x97t991751","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 71-87, May 2024. <br/> We consider a continuous-time homogeneous Markov process on the state space $mathbf{Z}_+={0,1,2,dots}$. The process is interpreted as the motion of a particle. A particle may transit only to neighboring points $mathbf{Z}_+$, i.e., for each single motion of the particle, its coordinate changes by 1. The process is equipped with a branching mechanism. Branching sources may be located at each point of $mathbf{Z}_+$. At a moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) by the same rules as the initial particle. To such a branching Markov process there corresponds a Jacobi matrix. In terms of orthogonal polynomials corresponding to this matrix, we obtain formulas for the mean number of particles at an arbitrary fixed point of $mathbf{Z}_+$ at time $t>0$. The results obtained are applied to some concrete models, an exact value for the mean number of particles in terms of special functions is given, and an asymptotic formula for this quantity for large time is found.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839539","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 : 2024-02-07DOI: 10.1137/s0040585x97t991635
H. Wakaki, V. V. Ulyanov
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 570-581, February 2024. We construct asymptotic expansions for the distribution function of the Bartlett--Nanda--Pillai statistic under the condition that the null linear hypothesis is valid in a multivariate linear model. Computable estimates of the accuracy of approximation are obtained via the Laplace approximation method, which is generalized to integrals for matrix-valued functions.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 570-581 页，2024 年 2 月。 在多元线性模型中零线性假设成立的条件下，我们构建了巴特利特-南达-皮莱统计量分布函数的渐近展开式。通过拉普拉斯近似法获得了近似精度的可计算估计值，并将其推广到矩阵值函数的积分。
{"title":"Laplace Expansion for Bartlett--Nanda--Pillai Test Statistic and Its Error Bound","authors":"H. Wakaki, V. V. Ulyanov","doi":"10.1137/s0040585x97t991635","DOIUrl":"https://doi.org/10.1137/s0040585x97t991635","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 570-581, February 2024. <br/> We construct asymptotic expansions for the distribution function of the Bartlett--Nanda--Pillai statistic under the condition that the null linear hypothesis is valid in a multivariate linear model. Computable estimates of the accuracy of approximation are obtained via the Laplace approximation method, which is generalized to integrals for matrix-valued functions.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769434","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 : 2024-02-07DOI: 10.1137/s0040585x97t991659
M. V. Zhitlukhin
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 607-621, February 2024. This paper is concerned with a dynamic game-theoretic model, where the players place bets on outcomes of random events or random vectors. Our purpose here is to construct a diffusion approximation of the model in the case where all players follow nearly optimal strategies. This approximation is further used to study the limit dynamics of the model.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 607-621 页，2024 年 2 月。 本文关注的是一个动态博弈论模型，其中玩家对随机事件或随机向量的结果下注。我们的目的是在所有玩家都遵循近乎最优策略的情况下，构建该模型的扩散近似值。这个近似值将进一步用于研究该模型的极限动态。
{"title":"On a Diffusion Approximation of a Prediction Game","authors":"M. V. Zhitlukhin","doi":"10.1137/s0040585x97t991659","DOIUrl":"https://doi.org/10.1137/s0040585x97t991659","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 607-621, February 2024. <br/> This paper is concerned with a dynamic game-theoretic model, where the players place bets on outcomes of random events or random vectors. Our purpose here is to construct a diffusion approximation of the model in the case where all players follow nearly optimal strategies. This approximation is further used to study the limit dynamics of the model.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770954","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 : 2024-02-07DOI: 10.1137/s0040585x97t991696
P. A. Yaskov
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 657-673, February 2024. We find general sufficient conditions in the Marchenko--Pastur theorem for high-dimensional sample covariance matrices associated with random vectors, for which the weak concentration property of quadratic forms may not hold in general. The results are obtained by means of a new martingale method, which may be useful in other problems of random matrix theory.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 657-673 页，2024 年 2 月。 我们为与随机向量相关的高维样本协方差矩阵找到了马琴科--帕斯图尔定理中的一般充分条件，对于这些矩阵，二次形式的弱集中特性可能在一般情况下不成立。这些结果是通过一种新的马氏方法得到的，这种方法可能对随机矩阵理论的其他问题有用。
{"title":"On Sufficient Conditions in the Marchenko--Pastur Theorem","authors":"P. A. Yaskov","doi":"10.1137/s0040585x97t991696","DOIUrl":"https://doi.org/10.1137/s0040585x97t991696","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 657-673, February 2024. <br/> We find general sufficient conditions in the Marchenko--Pastur theorem for high-dimensional sample covariance matrices associated with random vectors, for which the weak concentration property of quadratic forms may not hold in general. The results are obtained by means of a new martingale method, which may be useful in other problems of random matrix theory.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770969","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 : 2024-02-07DOI: 10.1137/s0040585x97t991684
E. A. Feinberg, A. N. Shiryaev
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 643-656, February 2024. In the present paper, we first give a survey of the forward and backward Kolmogorov equations for pure jump Markov processes with finite and countable state spaces, and then describe relevant results for the case of Markov processes with values in standard Borel spaces based on results of W. Feller and the authors of the present paper.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 643-656 页，2024 年 2 月。 在本文中，我们首先对具有有限和可数状态空间的纯跃迁马尔可夫过程的前向和后向科尔莫哥洛夫方程进行了考察，然后根据 W. Feller 和本文作者的结果，描述了在标准伯尔空间中具有值的马尔可夫过程情况下的相关结果。
{"title":"On Forward and Backward Kolmogorov Equations for Pure Jump Markov Processes and Their Generalizations","authors":"E. A. Feinberg, A. N. Shiryaev","doi":"10.1137/s0040585x97t991684","DOIUrl":"https://doi.org/10.1137/s0040585x97t991684","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 643-656, February 2024. <br/> In the present paper, we first give a survey of the forward and backward Kolmogorov equations for pure jump Markov processes with finite and countable state spaces, and then describe relevant results for the case of Markov processes with values in standard Borel spaces based on results of W. Feller and the authors of the present paper.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770961","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 : 2024-02-07DOI: 10.1137/s0040585x97t991611
V. I. Afanasyev
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 537-558, February 2024. A functional limit theorem is proved for a weakly supercritical branching process in a random environment under the condition that the process becomes extinct after time $nto infty $.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 537-558 页，2024 年 2 月。 证明了在随机环境中弱超临界分支过程在时间 $nto infty $ 之后灭绝的条件下的函数极限定理。
{"title":"Weakly Supercritical Branching Process in a Random Environment Dying at a Distant Moment","authors":"V. I. Afanasyev","doi":"10.1137/s0040585x97t991611","DOIUrl":"https://doi.org/10.1137/s0040585x97t991611","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 537-558, February 2024. <br/> A functional limit theorem is proved for a weakly supercritical branching process in a random environment under the condition that the process becomes extinct after time $nto infty $.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769299","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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