This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations.
{"title":"Proofs of the martingale FCLT","authors":"W. Whitt","doi":"10.1214/07-PS122","DOIUrl":"https://doi.org/10.1214/07-PS122","url":null,"abstract":"This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the ``martingale method'' \u0000for proving many-server heavy-traffic stochastic-process limits \u0000for queueing models, supporting diffusion-process approximations.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"4 1","pages":"268-302"},"PeriodicalIF":1.6,"publicationDate":"2007-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS122","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486652","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 formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.
{"title":"Differential equation approximations for Markov chains","authors":"R. Darling, J. Norris","doi":"10.1214/07-PS121","DOIUrl":"https://doi.org/10.1214/07-PS121","url":null,"abstract":"We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"37-79"},"PeriodicalIF":1.6,"publicationDate":"2007-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486410","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}
Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic inte- grals, martingale inequalities and limit theorems, likelihood-based methods in hypothesistesting and parameterestimation, and Studentizedpivots and bootstrap-t methods for confidence intervals. In contrast to standard nor- malization, large values of the observationsplay a lesser role as they appear both in the numerator and its self-normalized denominator, thereby mak- ing the process scale invariantand contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of de- pendentvariablesand describe a key method called "pseudo-maximization" that has been used to derive these results. In the multivariate case, self- normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case) and is ubiquitous in statistical applications, examples of which are given. AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35.
{"title":"Pseudo-maximization and self-normalized processes","authors":"V. Peña, M. Klass, T. Lai","doi":"10.1214/07-PS119","DOIUrl":"https://doi.org/10.1214/07-PS119","url":null,"abstract":"Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic inte- grals, martingale inequalities and limit theorems, likelihood-based methods in hypothesistesting and parameterestimation, and Studentizedpivots and bootstrap-t methods for confidence intervals. In contrast to standard nor- malization, large values of the observationsplay a lesser role as they appear both in the numerator and its self-normalized denominator, thereby mak- ing the process scale invariantand contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of de- pendentvariablesand describe a key method called \"pseudo-maximization\" that has been used to derive these results. In the multivariate case, self- normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case) and is ubiquitous in statistical applications, examples of which are given. AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"4 1","pages":"172-192"},"PeriodicalIF":1.6,"publicationDate":"2007-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS119","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486266","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}
In section 1, we present a number of classical results concerning the generalized Gamma convolution ( : GGC) variables, their Wiener-Gamma representations, and relation with Dirichlet processes. To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $Gamma_t(G)$
{"title":"Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples","authors":"Lancelot F. James, B. Roynette, M. Yor","doi":"10.1214/07-PS118","DOIUrl":"https://doi.org/10.1214/07-PS118","url":null,"abstract":"In section 1, we present a number of classical results concerning the generalized Gamma convolution ( : GGC) variables, their Wiener-Gamma representations, and relation with Dirichlet processes. To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $Gamma_t(G)$","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"346-415"},"PeriodicalIF":1.6,"publicationDate":"2007-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486106","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 survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright's constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.
{"title":"Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas","authors":"S. Janson","doi":"10.1214/07-PS104","DOIUrl":"https://doi.org/10.1214/07-PS104","url":null,"abstract":"This survey is a collection of various results and formulas by \u0000different authors \u0000on the areas (integrals) of five related processes, viz. Brownian \u0000motion, bridge, excursion, meander and double meander; \u0000for the Brownian motion and bridge, which take both positive and \u0000negative values, we consider both the integral of the absolute value \u0000and the integral of the positive (or negative) part. This gives us \u0000seven related positive random variables, for which we study, in particular, \u0000formulas for moments and Laplace transforms; we also give (in many \u0000cases) series \u0000representations and asymptotics for density functions and distribution \u0000functions. \u0000We further study Wright's constants arising in the asymptotic \u0000enumeration of connected graphs; \u0000these are known to be closely connected to the moments of the Brownian \u0000excursion area. \u0000 \u0000 \u0000The main purpose is to compare the results for these seven Brownian \u0000areas by stating the results in parallel forms; thus emphasizing both \u0000the similarities and the differences. \u0000A recurring theme is the Airy function which appears in slightly \u0000different ways in formulas for all seven random variables. \u0000We further want to \u0000give explicit relations between the many different \u0000similar notations and definitions that have been used by various \u0000authors. \u0000There are also some new results, mainly to fill in gaps left in the \u0000literature. Some short proofs are given, but most proofs are omitted \u0000and the reader is instead referred \u0000to the original sources.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"4 1","pages":"80-145"},"PeriodicalIF":1.6,"publicationDate":"2007-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486251","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 general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site. We allow for jump events at a site that affect the state of its neighbours. We give a law of large numbers and functional central limit theorem for additive set functions taken over an increasing family of subcubes of Z d . We discuss application to marked spatial point processes with births, deaths and jumps of particles, in particular examples such as continuum and lattice ballistic deposition and a sequential model for random loose sphere packing.
{"title":"Existence and spatial limit theorems for lattice and continuum particle systems","authors":"M. Penrose","doi":"10.1214/07-PS112","DOIUrl":"https://doi.org/10.1214/07-PS112","url":null,"abstract":"We give a general existence result for interacting particle \u0000systems with local interactions and bounded jump rates but \u0000noncompact state space at each site. We allow for \u0000jump events at a site that affect the state of \u0000its neighbours. We give a law of large \u0000numbers and functional central limit \u0000theorem for additive set functions taken over an increasing \u0000family of subcubes of Z d . We discuss application to \u0000marked spatial point processes with births, deaths and jumps of \u0000particles, in particular examples such as continuum and lattice ballistic \u0000deposition and a sequential model for random loose sphere packing.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"5 1","pages":"1-36"},"PeriodicalIF":1.6,"publicationDate":"2007-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS112","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486374","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 collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in sam- ples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with re- lated random variables, and draw connections with regular variation, which appears in various manifestations. AMS 2000 subject classifications: Primary 60F05, 60F15; secondary
{"title":"Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗","authors":"A. Gnedin, Ben B. Hansen, J. Pitman","doi":"10.1214/07-PS092","DOIUrl":"https://doi.org/10.1214/07-PS092","url":null,"abstract":"This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in sam- ples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with re- lated random variables, and draw connections with regular variation, which appears in various manifestations. AMS 2000 subject classifications: Primary 60F05, 60F15; secondary","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"4 1","pages":"146-171"},"PeriodicalIF":1.6,"publicationDate":"2007-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS092","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66485538","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}
Pub Date : 2007-01-08DOI: 10.1214/154957807000000013
A. Lejay
This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.
{"title":"On the constructions of the skew Brownian motion","authors":"A. Lejay","doi":"10.1214/154957807000000013","DOIUrl":"https://doi.org/10.1214/154957807000000013","url":null,"abstract":"This article summarizes the \u0000various ways one may use to construct \u0000the Skew Brownian motion, and shows their connections. \u0000Recent applications of this process in modelling and numerical \u0000simulation motivates this survey. \u0000This article ends with a brief account of related results, \u0000extensions and applications of the Skew Brownian motion.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"3 1","pages":"413-466"},"PeriodicalIF":1.6,"publicationDate":"2007-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/154957807000000013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66030820","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}
Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today's research in probability theory is `random media', including the now classical problems on `Brownian motion among obstacles' and the more recent `random walks in random environment' and `catalytic branching' models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in deterministic environments (sections 2-6) and in random media (sections 7-11).
{"title":"Branching diffusions, superdiffusions and random media","authors":"J. Engländer","doi":"10.1214/07-PS120","DOIUrl":"https://doi.org/10.1214/07-PS120","url":null,"abstract":"Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today's research in probability theory is `random media', including the now classical problems on `Brownian motion among obstacles' and the more recent `random walks in random environment' and `catalytic branching' models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in deterministic environments (sections 2-6) and in random media (sections 7-11).","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"4 1","pages":"303-364"},"PeriodicalIF":1.6,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS120","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66486288","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}
Pub Date : 2006-12-28DOI: 10.1214/154957806000000096
O. Haggstrom, J. Jonasson
This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${mathbb{Z}}^d$ and, more generally, on transitive graphs. For iid percolation on ${mathbb{Z}}^d$, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
{"title":"Uniqueness and Non-uniqueness in Percolation Theory","authors":"O. Haggstrom, J. Jonasson","doi":"10.1214/154957806000000096","DOIUrl":"https://doi.org/10.1214/154957806000000096","url":null,"abstract":"This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${mathbb{Z}}^d$ and, more generally, on transitive graphs. For iid percolation on ${mathbb{Z}}^d$, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"3 1","pages":"289-344"},"PeriodicalIF":1.6,"publicationDate":"2006-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/154957806000000096","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66030382","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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