The paper reviews existing results about the statistical distribution of zeros for three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of the main results.
{"title":"Statistical properties of zeta functions’ zeros","authors":"V. Kargin","doi":"10.1214/13-PS214","DOIUrl":"https://doi.org/10.1214/13-PS214","url":null,"abstract":"The paper reviews existing results about the statistical distribution of zeros for three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of the main results.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"11 1","pages":"121-160"},"PeriodicalIF":1.6,"publicationDate":"2013-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66002018","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}
ni=1 Xi and V 2 n = P n i=1 X 2 i . This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum Sn/Vn. Other self-normalized limit theorems are also briefly discussed. MSC 2010 subject classifications: Primary 60F05, 60F17; secondary 62E20.
{"title":"Self-normalized limit theorems: A survey","authors":"Q. Shao, Qiying Wang","doi":"10.1214/13-PS216","DOIUrl":"https://doi.org/10.1214/13-PS216","url":null,"abstract":"ni=1 Xi and V 2 n = P n i=1 X 2 i . This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum Sn/Vn. Other self-normalized limit theorems are also briefly discussed. MSC 2010 subject classifications: Primary 60F05, 60F17; secondary 62E20.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"10 1","pages":"69-93"},"PeriodicalIF":1.6,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/13-PS216","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66001577","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 review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.
{"title":"Distribution of the sum-of-digits function of random integers: A survey","authors":"Louis H. Y. Chen, Hsien-Kuei Hwang, V. Zacharovas","doi":"10.1214/12-PS213","DOIUrl":"https://doi.org/10.1214/12-PS213","url":null,"abstract":"We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"16 1","pages":"177-236"},"PeriodicalIF":1.6,"publicationDate":"2012-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/12-PS213","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65981750","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}
Author(s): Aldous, D; Lanoue, D | Abstract: To interpret interacting particle system style models as social dynamics, suppose each pair {i, j} of individuals in a finite population meet at random times of arbitrary specified rates vij, and update theirstates according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values Xi(t-),Xj (t-) arereplaced by 1/2 (Xi(t-) + Xj (t-)), 1/2 (Xi(t-) + Xj (t-)). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.
{"title":"A lecture on the averaging process","authors":"D. Aldous, D. Lanoue","doi":"10.1214/11-PS184","DOIUrl":"https://doi.org/10.1214/11-PS184","url":null,"abstract":"Author(s): Aldous, D; Lanoue, D | Abstract: To interpret interacting particle system style models as social dynamics, suppose each pair {i, j} of individuals in a finite population meet at random times of arbitrary specified rates vij, and update theirstates according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values Xi(t-),Xj (t-) arereplaced by 1/2 (Xi(t-) + Xj (t-)), 1/2 (Xi(t-) + Xj (t-)). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"90-102"},"PeriodicalIF":1.6,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS184","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965944","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}
An error is identified and corrected in the survey entitled 'Three theorems in discrete random geometry', published in Probability Surveys 8 (2011) 403-441. AMS 2000 subject classifications: Primary 60K35; secondary 82B43.
{"title":"Erratum: Three theorems in discrete random geometry","authors":"G. Grimmett","doi":"10.1214/12-PS210","DOIUrl":"https://doi.org/10.1214/12-PS210","url":null,"abstract":"An error is identified and corrected in the survey entitled 'Three theorems in discrete random geometry', published in Probability Surveys 8 (2011) 403-441. AMS 2000 subject classifications: Primary 60K35; secondary 82B43.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"438-438"},"PeriodicalIF":1.6,"publicationDate":"2012-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/12-PS210","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65982176","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 provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.
{"title":"On the notion(s) of duality for Markov processes","authors":"S. Jansen, N. Kurt","doi":"10.1214/12-PS206","DOIUrl":"https://doi.org/10.1214/12-PS206","url":null,"abstract":"We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"59-120"},"PeriodicalIF":1.6,"publicationDate":"2012-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/12-PS206","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65981584","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}
Following the recent survey by the same author of Szego's theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).
{"title":"Multivariate prediction and matrix Szego theory","authors":"N. Bingham","doi":"10.1214/12-PS200","DOIUrl":"https://doi.org/10.1214/12-PS200","url":null,"abstract":"Following the recent survey by the same author of Szego's theorem and \u0000orthogonal polynomials on the unit circle (OPUC) in the scalar case, we \u0000survey the corresponding multivariate prediction theory and matrix OPUC \u0000(MOPUC).","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"325-339"},"PeriodicalIF":1.6,"publicationDate":"2012-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65981449","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 present a list of equivalent expressions and extensions of Bougerol's celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.
{"title":"Bougerol’s identity in law and extensions","authors":"S. Vakeroudis","doi":"10.1214/12-PS195","DOIUrl":"https://doi.org/10.1214/12-PS195","url":null,"abstract":"We present a list of equivalent expressions and extensions of Bougerol's celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"411-437"},"PeriodicalIF":1.6,"publicationDate":"2012-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65981351","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}
Any negative moment of an increasing Lamperti process (Xt ; t 0) is a completely monotone function of t . This property enticed us to study systematically, for a given Markov process (Yt ; t 0) , the functions f such that the expectation of f(Yt) is a completely monotone function of t . We call these functions temporally completely monotone (for Y ). Our description of these functions is deduced from the analysis made by Ben Saad and Janen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a L evy process, a Bessel process, or an increasing Lamperti process.
{"title":"On temporally completely monotone functions for Markov processes","authors":"F. Hirsch, M. Yor","doi":"10.1214/11-PS179","DOIUrl":"https://doi.org/10.1214/11-PS179","url":null,"abstract":"Any negative moment of an increasing Lamperti process (Xt ; t 0) is a completely monotone function of t . This property enticed us to study systematically, for a given Markov process (Yt ; t 0) , the functions f such that the expectation of f(Yt) is a completely monotone function of t . We call these functions temporally completely monotone (for Y ). Our description of these functions is deduced from the analysis made by Ben Saad and Janen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a L evy process, a Bessel process, or an increasing Lamperti process.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"253-286"},"PeriodicalIF":1.6,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965675","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 is partly an expository paper. We prove and highlight a quantile inequality that is implicit in the fundamental paper by Komlos, Major, and Tusnady (31) on Brownian motion strong approximations to partial sums of independent and identically distributed random variables. We also derive a number of refinements of this inequality, which hold when more assumptions are added. A number of examples are detailed that will likely be of separate interest. We especially call attention to applications to the asymptotic equivalence theory of nonparametric statistical models and nonparametric function estimation. AMS 2000 subject classifications: Primary 62E17; secondary 62B15,
{"title":"Quantile coupling inequalities and their applications","authors":"D. Mason, Harrison H. Zhou","doi":"10.1214/12-PS198","DOIUrl":"https://doi.org/10.1214/12-PS198","url":null,"abstract":"This is partly an expository paper. We prove and highlight a quantile inequality that is implicit in the fundamental paper by Komlos, Major, and Tusnady (31) on Brownian motion strong approximations to partial sums of independent and identically distributed random variables. We also derive a number of refinements of this inequality, which hold when more assumptions are added. A number of examples are detailed that will likely be of separate interest. We especially call attention to applications to the asymptotic equivalence theory of nonparametric statistical models and nonparametric function estimation. AMS 2000 subject classifications: Primary 62E17; secondary 62B15,","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"439-479"},"PeriodicalIF":1.6,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65981389","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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