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On Symmetrized Chi-Square Tests in Autoregression with Outliers in Data 关于数据中存在异常值的自回归中的对称齐次方检验
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-02-07 DOI: 10.1137/s0040585x97t991623
M. V. Boldin
{"title":"On Symmetrized Chi-Square Tests in Autoregression with Outliers in Data","authors":"M. V. Boldin","doi":"10.1137/s0040585x97t991623","DOIUrl":"https://doi.org/10.1137/s0040585x97t991623","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 68, Issue 4, Page 559-569, February 2024. <br/> 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 $boldsymbol H_{Phi}colon G (x)in {Phi(x/theta),,theta&gt;0}$, 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 $gamma=0$.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":"16 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769433","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}
引用次数: 0
On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation 论正态逼近下精确渐近的矩完全收敛性
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-02-07 DOI: 10.1137/s0040585x97t991660
L. V. Rozovsky
{"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 &amp;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) $s+1 geqslant 0$, $rgeqslant 0$, $gamma&gt;1/2$, and $varepsilon&gt;0$, 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 $mathbf{E}|X|^{2+r/2}&lt;infty$ if $r &lt; 4$, $mathbf{E}|X|^r&lt;infty$ if $r&gt;4$, and $mathbf{E} X^4 ln{(1+|X|)}&lt;infty$ if $r=4$. Here, $xi$ is a standard Gaussian r.v.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":"20 1","pages":""},"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}
引用次数: 0
Kolmogorov's Last Discovery? (Kolmogorov and Algorithmic Statistics) 科尔莫戈罗夫的最后发现？(柯尔莫哥洛夫与算法统计）
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-02-07 DOI: 10.1137/s0040585x97t991647
A. L. Semenov, A. Kh. Shen, N. K. Vereshchagin
{"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 &amp;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":"16 1","pages":""},"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}
引用次数: 0
One Limit Theorem for Branching Random Walks 分支随机游走的一极限定理
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-02-07 DOI: 10.1137/s0040585x97t991672
N. V. Smorodina, E. B. Yarovaya
{"title":"One Limit Theorem for Branching Random Walks","authors":"N. V. Smorodina, E. B. Yarovaya","doi":"10.1137/s0040585x97t991672","DOIUrl":"https://doi.org/10.1137/s0040585x97t991672","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 68, Issue 4, Page 630-642, February 2024. <br/> 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 $ttoinfty$.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":"145 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771145","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}
引用次数: 0
Abstracts of Talks Given at the 8th International Conference on Stochastic Methods 第八届随机方法国际会议演讲摘要
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-02-07 DOI: 10.1137/s0040585x97t991702
A. N. Shiryaev
{"title":"Abstracts of Talks Given at the 8th International Conference on Stochastic Methods","authors":"A. N. Shiryaev","doi":"10.1137/s0040585x97t991702","DOIUrl":"https://doi.org/10.1137/s0040585x97t991702","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 68, Issue 4, Page 674-711, February 2024. <br/> 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.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":"20 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770956","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}
引用次数: 0
Wiener Spiral for Volatility Modeling 维纳螺旋波动模型
4区 数学
Theory of Probability and its Applications Pub Date : 2023-11-01 DOI: 10.1137/s0040585x97t991581
M. Fukasawa
{"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":"37 5","pages":"0"},"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}
引用次数: 0
On One Family of Random Operators 关于一类随机算子
4区 数学
Theory of Probability and its Applications Pub Date : 2023-11-01 DOI: 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":"38 2","pages":"0"},"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}
引用次数: 0
Kolmogorov Problems on Equations for Stationary and Transition Probabilities of Diffusion Processes 扩散过程平稳和转移概率方程的Kolmogorov问题
4区 数学
Theory of Probability and its Applications Pub Date : 2023-11-01 DOI: 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":"75 6","pages":"0"},"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}
引用次数: 0
On Characterization of Quantum Gaussian Measurement Channels 量子高斯测量信道的表征
4区 数学
Theory of Probability and its Applications Pub Date : 2023-11-01 DOI: 10.1137/s0040585x97t99157x
A. S. Holevo
{"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":"74 4","pages":"0"},"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}
引用次数: 0
The Kolmogorov Inequality for the Maximum of the Sum of Random Variables and Its Martingale Analogues 随机变量和最大值的Kolmogorov不等式及其鞅类似物
4区 数学
Theory of Probability and its Applications Pub Date : 2023-11-01 DOI: 10.1137/s0040585x97t991568
N. E. Kordzakhia, A. A. Novikov, A. N. Shiryaev
{"title":"The Kolmogorov Inequality for the Maximum of the Sum of Random Variables and Its Martingale Analogues","authors":"N. E. Kordzakhia, A. A. Novikov, A. N. Shiryaev","doi":"10.1137/s0040585x97t991568","DOIUrl":"https://doi.org/10.1137/s0040585x97t991568","url":null,"abstract":"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.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":"72 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135515948","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}
引用次数: 0
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