We present a synthesis of a number of developments which have been made around the celebrated Tsirelson’s equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $G$, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $G=mathbb{T}$.
{"title":"Around Tsirelson's equation, or: The evolution process may not explain everything","authors":"K. Yano, M. Yor","doi":"10.1214/15-PS256","DOIUrl":"https://doi.org/10.1214/15-PS256","url":null,"abstract":"We present a synthesis of a number of developments which have been made around the celebrated Tsirelson’s equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $G$, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $G=mathbb{T}$.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"12 1","pages":"1-12"},"PeriodicalIF":1.6,"publicationDate":"2009-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66046602","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 give a survey and unified treatment of functional inte- gral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self- avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an in- troduction to fermionic integrals. The fermionic integrals are in terms of anticommutingGrassmann variables,which can be convenientlyinterpreted as differential forms.
{"title":"Functional integral representations for self-avoiding walk ∗","authors":"D. Brydges, J. Imbrie, G. Slade","doi":"10.1214/09-PS152","DOIUrl":"https://doi.org/10.1214/09-PS152","url":null,"abstract":"We give a survey and unified treatment of functional inte- gral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self- avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an in- troduction to fermionic integrals. The fermionic integrals are in terms of anticommutingGrassmann variables,which can be convenientlyinterpreted as differential forms.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"6 1","pages":"34-61"},"PeriodicalIF":1.6,"publicationDate":"2009-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/09-PS152","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65932958","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}
Kingman’s theory of partition structures relates, via a natural sampling procedure, finite partitions to hypothetical infinite populations. Explicit formulas for distributions of such partitions are rare, the most notable exception being the Ewens sampling formula, and its two-parameter extension by Pitman. When one adds an extra structure to the partitions like a linear order on the set of blocks and regenerative properties, some representation theorems allow to get more precise information on the distribution. In these notes we survey recent developments of the theory of regenerative partitions and compositions. In particular, we discuss connection between ordered and unordered structures, regenerative properties of the Ewens-Pitman partitions, and asymptotics of the number of components.
{"title":"Regeneration in random combinatorial structures","authors":"A. Gnedin","doi":"10.1214/10-PS163","DOIUrl":"https://doi.org/10.1214/10-PS163","url":null,"abstract":"Kingman’s theory of partition structures relates, via a natural \u0000sampling procedure, finite partitions to hypothetical infinite populations. \u0000Explicit formulas for distributions of such partitions are rare, the most notable exception being the Ewens sampling formula, and its two-parameter \u0000extension by Pitman. When one adds an extra structure to the partitions \u0000like a linear order on the set of blocks and regenerative properties, some \u0000representation theorems allow to get more precise information on the distribution. In these notes we survey recent developments of the theory of \u0000regenerative partitions and compositions. In particular, we discuss connection between ordered and unordered structures, regenerative properties of \u0000the Ewens-Pitman partitions, and asymptotics of the number of components.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"7 1","pages":"105-156"},"PeriodicalIF":1.6,"publicationDate":"2009-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/10-PS163","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65948867","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}
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.
{"title":"Stochastic analysis of Bernoulli processes","authors":"Nicolas Privault","doi":"10.1214/08-PS139","DOIUrl":"https://doi.org/10.1214/08-PS139","url":null,"abstract":"These notes survey some aspects of discrete-time chaotic calculus \u0000 and its applications, based on the chaos representation property \u0000 for i.i.d. sequences of random variables. \u0000 The topics covered include the Clark formula and predictable \u0000 representation, anticipating calculus, covariance identities and \u0000 functional inequalities (such as deviation and logarithmic Sobolev \u0000 inequalities), and an application to option hedging in discrete time.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"435-483"},"PeriodicalIF":1.6,"publicationDate":"2008-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66508921","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}
Queueing networks constitute a large family of stochastic models, involving jobs that enter a network, compete for service, and eventually leave the network upon completion of service. Since the early 1990s, substantial attention has been devoted to the question of when such networks are stable. This monograph presents a summary of such work. Emphasis is placed on the use of fluid models in showing stability, and on examples of queueing networks that are unstable even when the arrival rate is less than the service rate. The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006, and is also being published in the Springer Lecture Notes series.
{"title":"Stability of queueing networks","authors":"M. Bramson","doi":"10.1214/08-PS137","DOIUrl":"https://doi.org/10.1214/08-PS137","url":null,"abstract":"Queueing networks constitute a large family of stochastic models, involving jobs that enter a network, compete for service, and eventually leave the network upon completion of service. Since the early 1990s, substantial attention has been devoted to the question of when such networks are stable. \u0000 \u0000This monograph presents a summary of such work. Emphasis is placed on the use of fluid models in showing stability, and on examples of queueing networks that are unstable even when the arrival rate is less than the service rate. \u0000 \u0000The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006, and is also being published in the Springer Lecture Notes series.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"169-345"},"PeriodicalIF":1.6,"publicationDate":"2008-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66508828","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}
A rather general risk model compounded by a stochastic return process is presented, together with integral–differential equations for the ruin probability. Exact solutions, numerical solutions, and asymptotic properties are discussed. Keywords: ruin probability; linear stochastic differential equation; Volterra integral-differential equation; numerical methods; asymptotics
{"title":"Ruin Models with Investment Income","authors":"J. Paulsen","doi":"10.1214/08-PS134","DOIUrl":"https://doi.org/10.1214/08-PS134","url":null,"abstract":"A rather general risk model compounded by a stochastic return process is presented, together with integral–differential equations for the ruin probability. Exact solutions, numerical solutions, and asymptotic properties are discussed. \u0000 \u0000 \u0000Keywords: \u0000 \u0000ruin probability; \u0000linear stochastic differential equation; \u0000Volterra integral-differential equation; \u0000numerical methods; \u0000asymptotics","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2008-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/08-PS134","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66509088","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 give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.
{"title":"The notion of ψ -weak dependence and its applications to bootstrapping time series","authors":"P. Doukhan, Michael H. Neumann","doi":"10.1214/06-PS086","DOIUrl":"https://doi.org/10.1214/06-PS086","url":null,"abstract":"We give an introduction to a notion of weak dependence which \u0000is more general than mixing and allows to treat for example processes driven \u0000by discrete innovations as they appear with time series bootstrap. As a \u0000typical example, we analyze autoregressive processes and their bootstrap \u0000analogues in detail and show how weak dependence can be easily derived \u0000from a contraction property of the process. Furthermore, we provide an \u0000overview of classes of processes possessing the property of weak dependence \u0000and describe important probabilistic results under such an assumption.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"146-168"},"PeriodicalIF":1.6,"publicationDate":"2008-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/06-PS086","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66477137","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 uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of spectrally positive Levy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.
{"title":"Proof(s) of the Lamperti representation of Continuous-State Branching Processes","authors":"M. Caballero, A. Lambert, Gerónimo Uribe Bravo","doi":"10.1214/09-PS154","DOIUrl":"https://doi.org/10.1214/09-PS154","url":null,"abstract":"This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of spectrally positive Levy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"6 1","pages":"62-89"},"PeriodicalIF":1.6,"publicationDate":"2008-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65933337","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}
De Finetti’s classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1, 2, 3], Hoover [41, 42], Kallenberg [44] and Kingman [47]. On the other hand, such exchangeable laws were first related to questions from combinatorics in an independent analysis by Fremlin and Talagrand [29], and again more recently in Tao [62], where they appear as a natural proxy for the ‘leading order statistics’ of colourings of large graphs or hypergraphs. Moreover, this relation appears implicitly in the study of various more bespoke formalisms for handling ‘limit objects’ of sequences of dense graphs or hypergraphs in a number of recent works, including Lovasz and Szegedy [52], Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi [17], Elek and Szegedy [24] and Razborov [54, 55]. However, the connection between these works and the earlier probabilistic structural results seems to have gone largely unappreciated. In this survey we recall the basic results of the theory of exchangeable laws, and then explain the probabilistic versions of various interesting questions from graph and hypergraph theory that their connection motivates (particularly extremal questions on the testability of properties for graphs and hypergraphs). We also locate the notions of exchangeability of interest to us in the context of other classes of probability measures subject to various symmetries, in particular contrasting the methods employed to analyze exchangeable laws with related structural results in ergodic theory, particular the Furstenberg-Zimmer structure theorem for probability-preserving ℤ-systems, which underpins Furstenberg’s ergodic-theoretic proof of Szemeredi’s Theorem. The forthcoming paper [10] will make a much more elaborate appeal to the link between exchangeable laws and dense (directed) hypergraphs to establish various results in property testing.
{"title":"On exchangeable random variables and the statistics of large graphs and hypergraphs","authors":"Tim Austin","doi":"10.1214/08-PS124","DOIUrl":"https://doi.org/10.1214/08-PS124","url":null,"abstract":"De Finetti’s classical result of [18] identifying the law of an \u0000exchangeable family of random variables as a mixture of i.i.d. laws was \u0000extended to structure theorems for more complex notions of exchangeability \u0000by Aldous [1, 2, 3], Hoover [41, 42], Kallenberg [44] and Kingman [47]. On \u0000the other hand, such exchangeable laws were first related to questions from \u0000combinatorics in an independent analysis by Fremlin and Talagrand [29], \u0000and again more recently in Tao [62], where they appear as a natural proxy \u0000for the ‘leading order statistics’ of colourings of large graphs or hypergraphs. Moreover, this relation appears implicitly in the study of various \u0000more bespoke formalisms for handling ‘limit objects’ of sequences of dense \u0000graphs or hypergraphs in a number of recent works, including Lovasz and \u0000Szegedy [52], Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi [17], \u0000Elek and Szegedy [24] and Razborov [54, 55]. However, the connection between these works and the earlier probabilistic structural results seems to \u0000have gone largely unappreciated. \u0000 \u0000In this survey we recall the basic results of the theory of exchangeable \u0000laws, and then explain the probabilistic versions of various interesting questions from graph and hypergraph theory that their connection motivates \u0000(particularly extremal questions on the testability of properties for graphs \u0000and hypergraphs). \u0000 \u0000We also locate the notions of exchangeability of interest to us in the \u0000context of other classes of probability measures subject to various symmetries, in particular contrasting the methods employed to analyze exchangeable laws with related structural results in ergodic theory, particular the Furstenberg-Zimmer structure theorem for probability-preserving \u0000ℤ-systems, which underpins Furstenberg’s ergodic-theoretic proof of Szemeredi’s Theorem. \u0000 \u0000The forthcoming paper [10] will make a much more elaborate appeal to \u0000the link between exchangeable laws and dense (directed) hypergraphs to \u0000establish various results in property testing.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"80-145"},"PeriodicalIF":1.6,"publicationDate":"2008-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66508986","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 an expository review paper illustrating the "martin- gale method" for proving many-server heavy-traffic stochastic-process lim- its for queueing models, supporting diffusion-process approximations. Care- ful treatment is given to an elementary model - the classical infinite-server model M/M/1, but models with finitely many servers and customer aban- donment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate- 1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stop- ping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate se- quence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.
{"title":"Martingale proofs of many-server heavy-traffic limits for Markovian queues ∗","authors":"G. Pang, Rishi Talreja, W. Whitt","doi":"10.1214/06-PS091","DOIUrl":"https://doi.org/10.1214/06-PS091","url":null,"abstract":"This is an expository review paper illustrating the \"martin- gale method\" for proving many-server heavy-traffic stochastic-process lim- its for queueing models, supporting diffusion-process approximations. Care- ful treatment is given to an elementary model - the classical infinite-server model M/M/1, but models with finitely many servers and customer aban- donment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate- 1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stop- ping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate se- quence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"4 1","pages":"193-267"},"PeriodicalIF":1.6,"publicationDate":"2007-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/06-PS091","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66477174","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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