Pub Date : 2024-02-07DOI: 10.1137/s0040585x97t991623
M. V. Boldin
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 559-569, February 2024. A linear stationary model $mathrm{AR}(p)$ with unknown expectation, coefficients, and the distribution function of innovations $G(x)$ is considered. Autoregression observations contain gross errors (outliers, contaminations). The distribution of contaminations $Pi$ is unknown, their intensity is $gamma n^{-1/2}$ with unknown $gamma$, and $n$ is the number of observations. The main problem here (among others) is to test the hypothesis on the normality of innovations $boldsymbol H_{Phi}colon G (x)in {Phi(x/theta),,theta>0}$, where $Phi(x)$ is the distribution function of the normal law $boldsymbol N(0,1)$. In this setting, the previously constructed tests for autoregression with zero expectation do not apply. As an alternative, we propose special symmetrized chi-square type tests. Under the hypothesis and $gamma=0$, their asymptotic distribution is free. We study the asymptotic power under local alternatives in the form of the mixture $G(x)=A_{n,Phi}(x):=(1-n^{-1/2})Phi(x/theta_0)+n^{-1/2}H(x)$, where $H(x)$ is a distribution function, and $theta_0^2$ is the unknown variance of the innovations under $boldsymbol H_{Phi}$. The asymptotic qualitative robustness of the tests is established in terms of equicontinuity of the family of limit powers (as functions of $gamma$, $Pi,$ and $H(x)$) relative to $gamma$ at the point $gamma=0$.
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Pub Date : 2024-02-07DOI: 10.1137/s0040585x97t991660
L. V. Rozovsky
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 622-629, February 2024. For the sums of the form $overline I_s(varepsilon) = sum_{ngeqslant 1} n^{s-r/2}mathbf{E}|S_n|^r,mathbf I[|S_n|geqslant varepsilon,n^gamma]$, where $S_n = X_1 +dots + X_n$, $X_n$, $ngeqslant 1$, is a sequence of independent and identically distributed random variables (r.v.'s) $s+1 geqslant 0$, $rgeqslant 0$, $gamma>1/2$, and $varepsilon>0$, new results on their behavior are provided. As an example, we obtain the following generalization of Heyde's result [J. Appl. Probab., 12 (1975), pp. 173--175]: for any $rgeqslant 0$, $lim_{varepsilonsearrow 0}varepsilon^{2}sum_{ngeqslant 1} n^{-r/2} mathbf{E}|S_n|^r,mathbf I[|S_n|geqslant varepsilon, n] =mathbf{E} |xi|^{r+2}$ if and only if $mathbf{E} X=0$ and $mathbf{E} X^2=1$, and also $mathbf{E}|X|^{2+r/2}<infty$ if $r < 4$, $mathbf{E}|X|^r<infty$ if $r>4$, and $mathbf{E} X^4 ln{(1+|X|)}<infty$ if $r=4$. Here, $xi$ is a standard Gaussian r.v.
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 622-629, February 2024. 对于和的形式 $overline I_s(varepsilon) = sum_{ngeqslant 1} n^{s-r/2}mathbf{E}|S_n|^r,mathbf I[|S_n|geqslant varepsilon、n^gamma]$，其中 $S_n = X_1 +dots + X_n$，$X_n$，$ngeqslant 1$，是一串独立且同分布的随机变量（r.v.'s） $s+1 geqslant 0$、$rgeqslant 0$、$gamma>1/2$ 和 $/varepsilon>0$，提供了关于它们行为的新结果。例如，我们得到了海德结果的以下概括[J. Appl、12 (1975), pp. 173--175]: 对于任意 $rgeqslant 0$, $lim_{varepsilonsearrow 0}varepsilon^{2}sum_{ngeqslant 1} n^{-r/2} mathbf{E}|S_n|^r,mathbf I[|S_n|geqslant varepsilon, n] =mathbf{E}|xi|^{r+2}$ if and only if $mathbf{E}X=0$ 和 $mathbf{E}X^2=1$, and also $mathbf{E}|X|^{2+r/2}<infty$ if $r < 4$, $mathbf{E}|X|^r<infty$ if $r>4$, and $mathbf{E}X^4 ln{(1+|X|)}<infty$ （如果 $r=4$）。这里，$xi$ 是标准高斯r.v.
{"title":"On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation","authors":"L. V. Rozovsky","doi":"10.1137/s0040585x97t991660","DOIUrl":"https://doi.org/10.1137/s0040585x97t991660","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 622-629, February 2024. <br/> For the sums of the form $overline I_s(varepsilon) = sum_{ngeqslant 1} n^{s-r/2}mathbf{E}|S_n|^r,mathbf I[|S_n|geqslant varepsilon,n^gamma]$, where $S_n = X_1 +dots + X_n$, $X_n$, $ngeqslant 1$, is a sequence of independent and identically distributed random variables (r.v.'s) $s+1 geqslant 0$, $rgeqslant 0$, $gamma>1/2$, and $varepsilon>0$, new results on their behavior are provided. As an example, we obtain the following generalization of Heyde's result [J. Appl. Probab., 12 (1975), pp. 173--175]: for any $rgeqslant 0$, $lim_{varepsilonsearrow 0}varepsilon^{2}sum_{ngeqslant 1} n^{-r/2} mathbf{E}|S_n|^r,mathbf I[|S_n|geqslant varepsilon, n] =mathbf{E} |xi|^{r+2}$ if and only if $mathbf{E} X=0$ and $mathbf{E} X^2=1$, and also $mathbf{E}|X|^{2+r/2}<infty$ if $r < 4$, $mathbf{E}|X|^r<infty$ if $r>4$, and $mathbf{E} X^4 ln{(1+|X|)}<infty$ if $r=4$. Here, $xi$ is a standard Gaussian r.v.","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":"139770964","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/s0040585x97t991647
A. L. Semenov, A. Kh. Shen, N. K. Vereshchagin
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 582-606, February 2024. The definition of descriptional complexity of finite objects suggested by Kolmogorov and other authors in the mid-1960s is now well known. In addition, Kolmogorov pointed out some approaches to a more fine-grained classification of finite objects, such as the resource-bounded complexity (1965), structure function (1974), and the notion of $(alpha,beta)$-stochasticity (1981). Later it turned out that these approaches are essentially equivalent in that they define the same curve in different coordinates. In this survey, we try to follow the development of these ideas of Kolmogorov as well as similar ideas suggested independently by other authors.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 582-606 页，2024 年 2 月。 20 世纪 60 年代中期，科尔莫戈罗夫和其他学者提出了有限对象描述复杂性的定义，这一定义现已广为人知。此外，科尔莫哥罗夫还指出了一些对有限对象进行更精细分类的方法，如资源约束复杂性（1965）、结构函数（1974）和$(alpha,beta)$随机性（1981）的概念。后来发现，这些方法本质上是等价的，因为它们用不同的坐标定义了同一条曲线。在本研究中，我们试图跟踪科尔莫格罗夫的这些观点以及其他作者独立提出的类似观点的发展。
{"title":"Kolmogorov's Last Discovery? (Kolmogorov and Algorithmic Statistics)","authors":"A. L. Semenov, A. Kh. Shen, N. K. Vereshchagin","doi":"10.1137/s0040585x97t991647","DOIUrl":"https://doi.org/10.1137/s0040585x97t991647","url":null,"abstract":"Theory of Probability &Its Applications, Volume 68, Issue 4, Page 582-606, February 2024. <br/> The definition of descriptional complexity of finite objects suggested by Kolmogorov and other authors in the mid-1960s is now well known. In addition, Kolmogorov pointed out some approaches to a more fine-grained classification of finite objects, such as the resource-bounded complexity (1965), structure function (1974), and the notion of $(alpha,beta)$-stochasticity (1981). Later it turned out that these approaches are essentially equivalent in that they define the same curve in different coordinates. In this survey, we try to follow the development of these ideas of Kolmogorov as well as similar ideas suggested independently by other authors.","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":"139769495","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/s0040585x97t991702
A. N. Shiryaev
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 674-711, February 2024. This paper presents abstracts of talks given at the 8th International Conference on Stochastic Methods (ICSM-8), held June 1--8, 2023 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. This year's conference was dedicated to the 120th birthday of Andrei Nikolaevich Kolmogorov and was chaired by A. N. Shiryaev. Participants included leading scientists from Russia, Portugal, and Tadjikistan.
概率论及其应用》（Theory of Probability &Its Applications），第68卷第4期，第674-711页，2024年2月。 本文收录了2023年6月1日至8日在Divnomorskoe（Gelendzhik镇附近）顿河国立技术大学Raduga体育健身中心举行的第八届随机方法国际会议（ICSM-8）上发表的演讲摘要。今年的会议是为了纪念安德烈-尼古拉耶维奇-科尔莫戈罗夫诞辰120周年，由A. N. Shiryaev主持。与会者包括来自俄罗斯、葡萄牙和塔吉克斯坦的顶尖科学家。
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Pub Date : 2024-02-07DOI: 10.1137/s0040585x97t991672
N. V. Smorodina, E. B. Yarovaya
Theory of Probability &Its Applications, Volume 68, Issue 4, Page 630-642, February 2024. The foundations of the general theory of Markov random processes were laid by A.N. Kolmogorov. Such processes include, in particular, branching random walks on lattices $mathbf{Z}^d$, $d in mathbf{N}$. In the present paper, we consider a branching random walk where particles may die or produce descendants at any point of the lattice. Motion of each particle on $mathbf{Z}^d$ is described by a symmetric homogeneous irreducible random walk. It is assumed that the branching rate of particles at $x in mathbf{Z}^d$ tends to zero as $|x| to infty$, and that an additional condition on the parameters of the branching random walk, which gives that the mean population size of particles at each point $mathbf{Z}^d$ grows exponentially in time, is met. In this case, the walk generation operator in the right-hand side of the equation for the mean population size of particles undergoes a perturbation due to possible generation of particles at points $mathbf{Z}^d$. Equations of this kind with perturbation of the diffusion operator in $mathbf{R}^2$, which were considered by Kolmogorov, Petrovsky, and Piskunov in 1937, continue being studied using the theory of branching random walks on discrete structures. Under the above assumptions, we prove a limit theorem on mean-square convergence of the normalized number of particles at an arbitrary fixed point of the lattice as $ttoinfty$.
概率论及其应用》（Theory of Probability &Its Applications），第 68 卷第 4 期，第 630-642 页，2024 年 2 月。 马尔可夫随机过程一般理论的基础是由 A.N. 科尔莫哥罗德夫（A.N. Kolmogorov）奠定的。这类过程尤其包括网格 $mathbf{Z}^d$, $d in mathbf{N}$上的分支随机游走。在本文中，我们考虑的是一种分支随机行走，粒子可能会在网格的任意点死亡或产生后代。每个粒子在 $mathbf{Z}^d$ 上的运动都是由对称同质不可还原随机行走描述的。假设粒子在 $x in mathbf{Z}^d$ 处的分支率随着 $|x| to infty$ 趋于零，并且满足分支随机行走参数的一个附加条件，即粒子在每个点 $mathbf{Z}^d$ 的平均种群数量随时间呈指数增长。在这种情况下，由于粒子可能在 $mathbf{Z}^d$ 点产生，粒子平均种群数量方程右侧的行走生成算子会发生扰动。科尔莫戈罗夫、彼得罗夫斯基和皮斯库诺夫曾在 1937 年考虑过这种带有 $mathbf{R}^2$ 中扩散算子扰动的方程，现在我们仍在使用离散结构上的分支随机游走理论对其进行研究。在上述假设条件下，我们证明了网格任意定点处粒子数归一化为 $ttoinfty$ 的均方收敛极限定理。
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Pub Date : 2023-11-01DOI: 10.1137/s0040585x97t991581
M. Fukasawa
Focusing on a lognormal stochastic volatility model, we present an elementary introduction to rough volatility modeling for financial assets with some new findings.Keywordsfractional Brownian motionimplied volatilityleverage effect
{"title":"Wiener Spiral for Volatility Modeling","authors":"M. Fukasawa","doi":"10.1137/s0040585x97t991581","DOIUrl":"https://doi.org/10.1137/s0040585x97t991581","url":null,"abstract":"Focusing on a lognormal stochastic volatility model, we present an elementary introduction to rough volatility modeling for financial assets with some new findings.Keywordsfractional Brownian motionimplied volatilityleverage effect","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135510182","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 : 2023-11-01DOI: 10.1137/s0040585x97t991556
I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev
{"title":"On One Family of Random Operators","authors":"I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev","doi":"10.1137/s0040585x97t991556","DOIUrl":"https://doi.org/10.1137/s0040585x97t991556","url":null,"abstract":"","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135510178","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 : 2023-11-01DOI: 10.1137/s0040585x97t991507
V. I. Bogachev, M. Röckner, S. V. Shaposhnikov
{"title":"Kolmogorov Problems on Equations for Stationary and Transition Probabilities of Diffusion Processes","authors":"V. I. Bogachev, M. Röckner, S. V. Shaposhnikov","doi":"10.1137/s0040585x97t991507","DOIUrl":"https://doi.org/10.1137/s0040585x97t991507","url":null,"abstract":"","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135514512","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 : 2023-11-01DOI: 10.1137/s0040585x97t99157x
A. S. Holevo
We provide a characterization of measurement (quantum-classical) channels, which map Gaussian states to Gaussian probability distributions.Keywordsquantum measurement channelGaussian distributionoperator characteristic function
{"title":"On Characterization of Quantum Gaussian Measurement Channels","authors":"A. S. Holevo","doi":"10.1137/s0040585x97t99157x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99157x","url":null,"abstract":"We provide a characterization of measurement (quantum-classical) channels, which map Gaussian states to Gaussian probability distributions.Keywordsquantum measurement channelGaussian distributionoperator characteristic function","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135515941","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 : 2023-11-01DOI: 10.1137/s0040585x97t991568
N. E. Kordzakhia, A. A. Novikov, A. N. Shiryaev
We give a survey of the results related to extensions of the Kolmogorov inequality for the distribution of the absolute value of the maximum of the sum of centered independent random variables to the case of martingales considered at random stopping times.
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