The existence of the bifractional Brownian motion ${B_{H,K}}$ indexed by a sphere when $Kin (-infty ,1]setminus {0}$ and $Hin (0,1/2]$ is discussed, and the asymptotics of its excursion probability $mathbb{P}left{{sup _{Min mathbb{S}}}{B_{H,K}}(M)>xright}$ as $xto infty $ is studied.
{"title":"Notes on spherical bifractional Brownian motion","authors":"Mohamed El Omari","doi":"10.15559/22-vmsta207","DOIUrl":"https://doi.org/10.15559/22-vmsta207","url":null,"abstract":"The existence of the bifractional Brownian motion ${B_{H,K}}$ indexed by a sphere when $Kin (-infty ,1]setminus {0}$ and $Hin (0,1/2]$ is discussed, and the asymptotics of its excursion probability $mathbb{P}left{{sup _{Min mathbb{S}}}{B_{H,K}}(M)>xright}$ as $xto infty $ is studied.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"2013 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86201708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.
{"title":"Lévy processes conditioned to stay in a half-space with applications to directional extremes","authors":"J. Ivanovs, Jakob D. Thostesen","doi":"10.15559/22-vmsta217","DOIUrl":"https://doi.org/10.15559/22-vmsta217","url":null,"abstract":"This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"30 4","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72475543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.
{"title":"A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution","authors":"L. Bianchi, S. Bonaccorsi, L. Tubaro","doi":"10.15559/22-VMSTA216","DOIUrl":"https://doi.org/10.15559/22-VMSTA216","url":null,"abstract":"We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"103 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79464898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $check{text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.
{"title":"A limit theorem for persistence diagrams of random filtered complexes built over marked point processes","authors":"T. Shirai, K. Suzaki","doi":"10.15559/22-vmsta214","DOIUrl":"https://doi.org/10.15559/22-vmsta214","url":null,"abstract":"Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $check{text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"44 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87416529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-Gaussian and sub-gamma bounds previously studied in this context. The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.
{"title":"Bernstein-type bounds for beta distribution","authors":"M. Skorski","doi":"10.15559/23-vmsta223","DOIUrl":"https://doi.org/10.15559/23-vmsta223","url":null,"abstract":"This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-Gaussian and sub-gamma bounds previously studied in this context.\u0000The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"30 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83131000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bias reduction of a conditional maximum likelihood estimator for a Gaussian second-order moving average model","authors":"Fumiaki Honda, T. Kurosawa","doi":"10.15559/21-vmsta187","DOIUrl":"https://doi.org/10.15559/21-vmsta187","url":null,"abstract":"","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"108 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79961794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Principal components analysis for mixtures with varying concentrations","authors":"O. Sugakova, R. Maiboroda","doi":"10.15559/21-vmsta191","DOIUrl":"https://doi.org/10.15559/21-vmsta191","url":null,"abstract":"","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"37 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86285675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bounded in the mean solutions of a second-order difference equation","authors":"M. Horodnii, V. Kravets","doi":"10.15559/21-vmsta189","DOIUrl":"https://doi.org/10.15559/21-vmsta189","url":null,"abstract":"","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"252 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76315592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices","authors":"Y. Kawamoto","doi":"10.15559/21-vmsta193","DOIUrl":"https://doi.org/10.15559/21-vmsta193","url":null,"abstract":"","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"48 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74523612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Malliavin–Stein method: a survey of some recent developments","authors":"E. Azmoodeh, G. Peccati, Xiaochuan Yang","doi":"10.15559/21-vmsta184","DOIUrl":"https://doi.org/10.15559/21-vmsta184","url":null,"abstract":"","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"37 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84747600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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