Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991945
M. P. Savelov
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 322-330, August 2024. We obtain two-sided estimates for the weighted sum of probabilities of errors in the multiple hypothesis testing problem with finite number of hypotheses on a nonhomogeneous sample of size $n$. The obtained upper and lower estimates are shown to converge to zero exponentially fast with increasing $n$ in a wide class of cases. The results obtained can be used for deriving two-sided estimates for the size of a sample required for multiple hypothesis testing.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 2 期，第 322-330 页，2024 年 8 月。 我们获得了大小为 $n$ 的非均质样本上有限个假设的多重假设检验问题中错误概率加权和的双侧估计值。结果表明，在很多情况下，随着 $n$ 的增大，所得到的上估计值和下估计值会以指数级的速度趋近于零。所得结果可用于推导多重假设检验所需的样本大小的双侧估计值。
{"title":"Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample","authors":"M. P. Savelov","doi":"10.1137/s0040585x97t991945","DOIUrl":"https://doi.org/10.1137/s0040585x97t991945","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 322-330, August 2024. <br/> We obtain two-sided estimates for the weighted sum of probabilities of errors in the multiple hypothesis testing problem with finite number of hypotheses on a nonhomogeneous sample of size $n$. The obtained upper and lower estimates are shown to converge to zero exponentially fast with increasing $n$ in a wide class of cases. The results obtained can be used for deriving two-sided estimates for the size of a sample required for multiple hypothesis testing.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198604","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 : 2024-08-14DOI: 10.1137/s0040585x97t991842
P. K. Babilua, E. A. Nadaraya
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 173-185, August 2024. The limiting distribution of the integral square deviation of a nonparametric kernel-type estimator for the Poisson regression function is established. A criterion for testing the hypothesis on the Poisson regression function is constructed. The power asymptotic of the constructed criterion is studied for certain types of close alternatives.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 2 期，第 173-185 页，2024 年 8 月。 建立了泊松回归函数非参数核型估计器积分平方差的极限分布。构建了检验泊松回归函数假设的准则。针对某些类型的近似替代方案，研究了所构建标准的幂渐近性。
{"title":"On One Nonparametric Estimation of the Poisson Regression Function","authors":"P. K. Babilua, E. A. Nadaraya","doi":"10.1137/s0040585x97t991842","DOIUrl":"https://doi.org/10.1137/s0040585x97t991842","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 173-185, August 2024. <br/> The limiting distribution of the integral square deviation of a nonparametric kernel-type estimator for the Poisson regression function is established. A criterion for testing the hypothesis on the Poisson regression function is constructed. The power asymptotic of the constructed criterion is studied for certain types of close alternatives.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198636","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 : 2024-08-14DOI: 10.1137/s0040585x97t99191x
V. E. Mosyagin
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 281-293, August 2024. Consider the random process $Y(t)=at-nu_+(pt)+nu_-(-qt)$, $tin(-infty,infty)$, where $nu_{pm}(t)$ are independent standard Poisson processes for $tgeqslant 0$ and $nu_{pm}(t)=0$ for $t<0$. The parameters $a$, $p$, and $q$ are such that $mathbf{E}Y(t)<0$, $tneq0$. We evaluate the sums $varphi_m(z,r)=sum_{kgeq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,dots$, $zgeq0$, of function series with parameter $ rin(0,1) $. These series are used for recursive evaluation of the moments $mathbf{E}(t^*)^m$, $mgeq 1$, for the time $t^*$ when the trajectory of the process $Y(t)$ attains its maximum value. The results obtained are applied to the problem of estimating the parameter $theta$ from $n$ observations with density $f(x,theta)$, which has a jump at the point $x=x(theta)$, $x'(theta)neq 0$. If $widehattheta_n$ is a maximum likelihood estimator for the true parameter $theta_0$, then the limit distribution as $ntoinfty$ for the normalized estimators $n(widehattheta_n-theta_0)$ is the distribution of the argument of the maximum $t^*_{theta_0}$ of the trajectory of the process $Y(t)$ with parameters $a$, $p$, and $q$, which depend on both the one-sided limits of the density at the point $x(theta_0)$ and the derivative $x'(theta_0)$. In this case, by evaluating the moments $mathbf{E}(t^*_{theta_0})^m$, $m=1, 2$, one can estimate both the asymptotic bias for the maximum likelihood estimator and its efficiency.
{"title":"Poisson Process with Linear Drift and Related Function Series","authors":"V. E. Mosyagin","doi":"10.1137/s0040585x97t99191x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99191x","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 281-293, August 2024. <br/> Consider the random process $Y(t)=at-nu_+(pt)+nu_-(-qt)$, $tin(-infty,infty)$, where $nu_{pm}(t)$ are independent standard Poisson processes for $tgeqslant 0$ and $nu_{pm}(t)=0$ for $t<0$. The parameters $a$, $p$, and $q$ are such that $mathbf{E}Y(t)<0$, $tneq0$. We evaluate the sums $varphi_m(z,r)=sum_{kgeq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,dots$, $zgeq0$, of function series with parameter $ rin(0,1) $. These series are used for recursive evaluation of the moments $mathbf{E}(t^*)^m$, $mgeq 1$, for the time $t^*$ when the trajectory of the process $Y(t)$ attains its maximum value. The results obtained are applied to the problem of estimating the parameter $theta$ from $n$ observations with density $f(x,theta)$, which has a jump at the point $x=x(theta)$, $x'(theta)neq 0$. If $widehattheta_n$ is a maximum likelihood estimator for the true parameter $theta_0$, then the limit distribution as $ntoinfty$ for the normalized estimators $n(widehattheta_n-theta_0)$ is the distribution of the argument of the maximum $t^*_{theta_0}$ of the trajectory of the process $Y(t)$ with parameters $a$, $p$, and $q$, which depend on both the one-sided limits of the density at the point $x(theta_0)$ and the derivative $x'(theta_0)$. In this case, by evaluating the moments $mathbf{E}(t^*_{theta_0})^m$, $m=1, 2$, one can estimate both the asymptotic bias for the maximum likelihood estimator and its efficiency.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198505","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/s0040585x97t991775
A. V. Shklyaev
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 99-116, May 2024. Let ${Z_n,, nge 0}$ be a branching process in an independent and identically distributed (i.i.d.) random environment and ${S_n,, n,{ge}, 1}$ be the associated random walk with steps $xi_i$. Under the Cramér condition on $xi_1$ and moment assumptions on a number of descendants of one particle, we know the asymptotics of the large deviation probabilities $mathbf{P}(ln Z_n > x)$, where $x/n > mu^*$. Here, $mu^*$ is a parameter depending on the process type. We study the asymptotic behavior of the process trajectory under the condition of a large deviation event. In particular, we obtain a conditional functional limit theorem for the trajectory of $(Z_{[nt]},, tin [0,1])$ given $ln Z_n>x$. This result is obtained in a more general model of linear recurrence sequence $Y_{n+1}=A_n Y_n + B_n$, $nge 0$, where ${A_i}$ is a sequence of i.i.d. random variables, $Y_0$, $B_i$, $ige 0$, are possibly dependent and have different distributions, and we need only some moment assumptions on them.
{"title":"Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event","authors":"A. V. Shklyaev","doi":"10.1137/s0040585x97t991775","DOIUrl":"https://doi.org/10.1137/s0040585x97t991775","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 99-116, May 2024. <br/> Let ${Z_n,, nge 0}$ be a branching process in an independent and identically distributed (i.i.d.) random environment and ${S_n,, n,{ge}, 1}$ be the associated random walk with steps $xi_i$. Under the Cramér condition on $xi_1$ and moment assumptions on a number of descendants of one particle, we know the asymptotics of the large deviation probabilities $mathbf{P}(ln Z_n > x)$, where $x/n > mu^*$. Here, $mu^*$ is a parameter depending on the process type. We study the asymptotic behavior of the process trajectory under the condition of a large deviation event. In particular, we obtain a conditional functional limit theorem for the trajectory of $(Z_{[nt]},, tin [0,1])$ given $ln Z_n>x$. This result is obtained in a more general model of linear recurrence sequence $Y_{n+1}=A_n Y_n + B_n$, $nge 0$, where ${A_i}$ is a sequence of i.i.d. random variables, $Y_0$, $B_i$, $ige 0$, are possibly dependent and have different distributions, and we need only some moment assumptions on them.","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":"140839361","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/s0040585x97t99174x
M. A. Lifshits, S. E. Nikitin
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 59-70, May 2024. We consider an energy saving approximation of a Wiener process under unilateral constraints. We show that, almost surely, on large time intervals the minimal energy necessary for the approximation logarithmically depends on the interval length. We also construct an adaptive approximation strategy optimal in a class of diffusion strategies and providing the logarithmic order of energy consumption.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 1 期，第 59-70 页，2024 年 5 月。 我们考虑了单边约束下维纳过程的节能近似。我们证明，在大时间间隔内，近似所需的最小能量几乎肯定与时间间隔长度成对数关系。我们还构建了一类扩散策略中最优的自适应逼近策略，并提供了能量消耗的对数阶。
{"title":"Energy Saving Approximation of Wiener Process under Unilateral Constraints","authors":"M. A. Lifshits, S. E. Nikitin","doi":"10.1137/s0040585x97t99174x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99174x","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 59-70, May 2024. <br/> We consider an energy saving approximation of a Wiener process under unilateral constraints. We show that, almost surely, on large time intervals the minimal energy necessary for the approximation logarithmically depends on the interval length. We also construct an adaptive approximation strategy optimal in a class of diffusion strategies and providing the logarithmic order of energy consumption.","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":"140839357","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/s0040585x97t991799
Y. Dong, L. Vostrikova
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 127-149, May 2024. This article is devoted to maximization of HARA (hyperbolic absolute risk aversion) utilities of the exponential Lévy switching processes on a finite time interval via the dual method. The description of all $f$-divergence minimal martingale measures and the expression of their Radon--Nikodým densities involving the Hellinger and Kulback--Leibler processes are given. The optimal strategies in progressively enlarged filtration for the maximization of HARA utilities as well as the values of the corresponding maximal expected utilities are derived. As an example, the Brownian switching model is presented with financial interpretations of the results via the value process.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 1 期，第 127-149 页，2024 年 5 月。 本文致力于通过对偶法最大化有限时间间隔上指数莱维切换过程的 HARA（双曲绝对风险厌恶）效用。文中给出了涉及海灵格和库尔贝克--莱布勒过程的所有$f$-发散最小马廷式度量的描述及其拉顿-尼科戴姆密度的表达式。推导了在逐步扩大的过滤中实现 HARA 效用最大化的最优策略以及相应的最大期望效用值。以布朗转换模型为例，通过价值过程对结果进行了金融解释。
{"title":"Utility Maximization of the Exponential Lévy Switching Models","authors":"Y. Dong, L. Vostrikova","doi":"10.1137/s0040585x97t991799","DOIUrl":"https://doi.org/10.1137/s0040585x97t991799","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 127-149, May 2024. <br/> This article is devoted to maximization of HARA (hyperbolic absolute risk aversion) utilities of the exponential Lévy switching processes on a finite time interval via the dual method. The description of all $f$-divergence minimal martingale measures and the expression of their Radon--Nikodým densities involving the Hellinger and Kulback--Leibler processes are given. The optimal strategies in progressively enlarged filtration for the maximization of HARA utilities as well as the values of the corresponding maximal expected utilities are derived. As an example, the Brownian switching model is presented with financial interpretations of the results via the value process.","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":"140839352","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/s0040585x97t991817
A.N. Shiryaev, E.B. Yarkovaya, V.A. Kutsenko
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 161-165, May 2024. This paper presents summaries of talks given during the 2023 spring semester of the General Seminar of the Department of Probability, Moscow State University. The seminar was held under the direction of A. N. Kolmogorov and B. V. Gnedenko. Current information about the seminar is available at http://new.math.msu.su/department/probab/seminar.html.
概率论及其应用》（Theory of Probability &Its Applications），第69卷第1期，第161-165页，2024年5月。 本文摘要介绍了莫斯科国立大学概率论系 2023 年春季学期总研讨会的演讲内容。研讨会由 A. N. Kolmogorov 和 B. V. Gnedenko 指导。有关该研讨会的最新信息请访问 http://new.math.msu.su/department/probab/seminar.html。
{"title":"News of Scientific Life - Information on the General Seminar of the Department of Probability, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia, Spring Term 2023","authors":"A.N. Shiryaev, E.B. Yarkovaya, V.A. Kutsenko","doi":"10.1137/s0040585x97t991817","DOIUrl":"https://doi.org/10.1137/s0040585x97t991817","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 161-165, May 2024. <br/> This paper presents summaries of talks given during the 2023 spring semester of the General Seminar of the Department of Probability, Moscow State University. The seminar was held under the direction of A. N. Kolmogorov and B. V. Gnedenko. Current information about the seminar is available at http://new.math.msu.su/department/probab/seminar.html.","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":"140839547","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/s0040585x97t991714
D. A. Borzykh
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 1-24, May 2024. Let $Lambda$ be the set of all boundary joint laws $operatorname{Law} ([X_a, A_a], [X_b, A_b])$ at times $t=a$ and $t=b$ of integrable increasing processes $(X_t)_{t in [a, b]}$ and their compensators $(A_t)_{t in [a, b]}$, which start at the initial time from an arbitrary integrable initial condition $[X_a, A_a]$. We show that $Lambda$ is convex and closed relative to the $psi$-weak topology with linearly growing gauge function $psi$. We obtain necessary and sufficient conditions for a probability measure $lambda$ on $mathcal{B}(mathbf{R}^2 times mathbf{R}^2)$ to lie in the class of measures $Lambda$. The main result of the paper provides, for two measures $mu_a$ and $mu_b$ on $mathcal{B}(mathbf{R}^2)$, necessary and sufficient conditions for the set $Lambda$ to contain a measure $lambda$ for which $mu_a$ and $mu_b$ are marginal distributions.
概率论及其应用》第 69 卷第 1 期第 1-24 页，2024 年 5 月。 让 $Lambda$ 是可积分递增过程 $(X_t)_{t in [a、b]}$及其补偿器 $(A_t)_{t in [a, b]}$，它们在初始时刻从任意可积分初始条件 $[X_a, A_a]$ 开始。我们证明，相对于具有线性增长规函数 $psi$ 的 $psi$ 弱拓扑，$Lambda$ 是凸的和封闭的。我们得到了$mathcal{B}(mathbf{R}^2 times mathbf{R}^2)$上的概率度量$lambda$位于度量类$Lambda$中的必要条件和充分条件。本文的主要结果为$mathcal{B}(mathbf{R}^2)$上的两个度量$mu_a$和$mu_b$提供了集合$Lambda$包含一个度量$lambda$的必要条件和充分条件，其中$mu_a$和$mu_b$是边际分布。
{"title":"Joint Distributions of Generalized Integrable Increasing Processes and Their Generalized Compensators","authors":"D. A. Borzykh","doi":"10.1137/s0040585x97t991714","DOIUrl":"https://doi.org/10.1137/s0040585x97t991714","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 1-24, May 2024. <br/> Let $Lambda$ be the set of all boundary joint laws $operatorname{Law} ([X_a, A_a], [X_b, A_b])$ at times $t=a$ and $t=b$ of integrable increasing processes $(X_t)_{t in [a, b]}$ and their compensators $(A_t)_{t in [a, b]}$, which start at the initial time from an arbitrary integrable initial condition $[X_a, A_a]$. We show that $Lambda$ is convex and closed relative to the $psi$-weak topology with linearly growing gauge function $psi$. We obtain necessary and sufficient conditions for a probability measure $lambda$ on $mathcal{B}(mathbf{R}^2 times mathbf{R}^2)$ to lie in the class of measures $Lambda$. The main result of the paper provides, for two measures $mu_a$ and $mu_b$ on $mathcal{B}(mathbf{R}^2)$, necessary and sufficient conditions for the set $Lambda$ to contain a measure $lambda$ for which $mu_a$ and $mu_b$ are marginal distributions.","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":"140839312","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/s0040585x97t991787
A. L. Yakymiv
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 117-126, May 2024. We consider a random permutation $tau_n$ uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the so-called $A$-permutations). Let $zeta_n$ be the total number of cycles, and let $eta_n(1)leqeta_n(2)leqdotsleqeta_n(zeta_n)$ be the ordered sample of cycle lengths of the permutation $tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $eta_n(m)$ with numbers $m$ in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence $eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340--357] who considered the case $A=mathbf N$.
{"title":"Limit Behavior of Order Statistics on Cycle Lengths of Random $A$-Permutations","authors":"A. L. Yakymiv","doi":"10.1137/s0040585x97t991787","DOIUrl":"https://doi.org/10.1137/s0040585x97t991787","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 1, Page 117-126, May 2024. <br/> We consider a random permutation $tau_n$ uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the so-called $A$-permutations). Let $zeta_n$ be the total number of cycles, and let $eta_n(1)leqeta_n(2)leqdotsleqeta_n(zeta_n)$ be the ordered sample of cycle lengths of the permutation $tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $eta_n(m)$ with numbers $m$ in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence $eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340--357] who considered the case $A=mathbf N$.","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":"140839363","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/s0040585x97t991738
Yu. Yu. Linke, I. S. Borisov
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 35-58, May 2024. Let $f_1(t), dots, f_n(t)$ be independent copies of some a.s. continuous stochastic process $f(t)$, $tin[0,1]$, which are observed with noise. We consider the problem of nonparametric estimation of the mean function $mu(t) = mathbf{E}f(t)$ and of the covariance function $psi(t,s)=operatorname{Cov}{f(t),f(s)}$ if the noise values of each of the copies $f_i(t)$, $i=1,dots,n$, are observed in some collection of generally random (in general) time points (regressors). Under wide assumptions on the time points, we construct uniformly consistent kernel estimators for the mean and covariance functions both in the case of sparse data (where the number of observations for each copy of the stochastic process is uniformly bounded) and in the case of dense data (where the number of observations at each of $n$ series is increasing as $ntoinfty$). In contrast to the previous studies, our kernel estimators are universal with respect to the structure of time points, which can be either fixed rather than necessarily regular, or random rather than necessarily formed of independent or weakly dependent random variables.
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