Steven R. Howard, Aaditya Ramdas, Jon D. McAuliffe, J. Sekhon
We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960-80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980-2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Pe~na; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cram'er-Chernoff method, self-normalized processes, and other parts of the literature.
我们为鞅序列跨越时间相关线性阈值的概率建立了一类指数界。我们的主要见解是，以这种方式制定指数浓度不等式既自然又富有成效。我们通过提出一个单一的假设和定理来说明这一点，这些假设和定理统一并加强了鞅的许多尾界，包括Bernstein, Bennett, Hoeffding和Freedman的经典不等式(1960-80);当代不平等(1980-2000)，作者:Shorack和Wellner、Pinelis、Blackwell、van de Geer和de la Pe ~na;以及Khan、Tropp、Bercu、Touati、Delyon等人的一些现代不平等现象(2000年后)。在每一种情况下，我们给出了迄今为止最强大和最一般的陈述，量化了离散和连续时间中各种非参数假设下标量、矩阵和巴拿赫空间值鞅的时间均匀浓度。在这样做的过程中，我们弥合了现有的跨线不等式、序列概率比检验、Cram - chernoff方法、自归一化过程和其他部分文献之间的差距。
{"title":"Time-uniform Chernoff bounds via nonnegative supermartingales","authors":"Steven R. Howard, Aaditya Ramdas, Jon D. McAuliffe, J. Sekhon","doi":"10.1214/18-ps321","DOIUrl":"https://doi.org/10.1214/18-ps321","url":null,"abstract":"We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960-80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980-2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Pe~na; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cram'er-Chernoff method, self-normalized processes, and other parts of the literature.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46546439","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 recent results on the metastable behavior of continuous-time Markov chains derived through the characterization of Markov chains as unique solutions of martingale problems.
{"title":"Metastable Markov chains","authors":"C. Landim","doi":"10.1214/18-PS310","DOIUrl":"https://doi.org/10.1214/18-PS310","url":null,"abstract":"We review recent results on the metastable behavior of continuous-time Markov chains derived through the characterization of Markov chains as unique solutions of martingale problems.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48054704","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 explores the equivalences between four definitions of uniform large deviations principles and uniform Laplace principles found in the literature. Counterexamples are presented to illustrate the differences between these definitions and specific conditions are described under which these definitions are equivalent to each other. A fifth definition called the equicontinuous uniform Laplace principle (EULP) is proposed and proven to be equivalent to Freidlin and Wentzell's definition of a uniform large deviations principle. Sufficient conditions that imply a measurable function of infinite dimensional Wiener process satisfies an EULP using the variational methods of Budhiraja, Dupuis and Maroulas are presented. Finally, this theory is applied to prove that a family of Hilbert space valued stochastic equations exposed to multiplicative noise satisfy a uniform large deviations principle that is uniform over all initial conditions in bounded subsets of the Hilbert space.
{"title":"Equivalences and counterexamples between several definitions of the uniform large deviations principle","authors":"M. Salins","doi":"10.1214/18-PS309","DOIUrl":"https://doi.org/10.1214/18-PS309","url":null,"abstract":"This paper explores the equivalences between four definitions of uniform large deviations principles and uniform Laplace principles found in the literature. Counterexamples are presented to illustrate the differences between these definitions and specific conditions are described under which these definitions are equivalent to each other. A fifth definition called the equicontinuous uniform Laplace principle (EULP) is proposed and proven to be equivalent to Freidlin and Wentzell's definition of a uniform large deviations principle. Sufficient conditions that imply a measurable function of infinite dimensional Wiener process satisfies an EULP using the variational methods of Budhiraja, Dupuis and Maroulas are presented. Finally, this theory is applied to prove that a family of Hilbert space valued stochastic equations exposed to multiplicative noise satisfy a uniform large deviations principle that is uniform over all initial conditions in bounded subsets of the Hilbert space.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43075438","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 study a model of random $mathcal{R}$-enriched trees that is based on weights on the $mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when $mathcal{R}$ is a composite class, and a Gromov--Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the $mathcal{R}$-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini--Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest $2$-connected component in random graphs from planar-like classes. We prove Benjamini--Schramm convergence of random $k$-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their $2$-connected components.
{"title":"Limits of random tree-like discrete structures","authors":"Benedikt Stufler","doi":"10.1214/19-ps338","DOIUrl":"https://doi.org/10.1214/19-ps338","url":null,"abstract":"We study a model of random $mathcal{R}$-enriched trees that is based on weights on the $mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when $mathcal{R}$ is a composite class, and a Gromov--Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the $mathcal{R}$-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini--Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest $2$-connected component in random graphs from planar-like classes. We prove Benjamini--Schramm convergence of random $k$-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their $2$-connected components.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2016-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66079960","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}
H. Duminil-Copin, Maxime Gagnebin, Matan Harel, I. Manolescu, V. Tassion
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector (psi) and energy (Lambda), which satisfy (V psi = Lambda psi), where (V) is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights (a= b=1) and (c > 0). We also show that the same vector (psi) satisfies (H psi = E psi), where (H) is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value (E) computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on (mathbb{Z}^{2}) with cluster weight (q >4) exhibits a first-order phase transition.
在本文中，我们回顾了一些已知的关于贝氏坐标的事实。我们提出一个详细的建设是拟设向量(psi)和能源(λ),满足(Vψ=λpsi), (V)在哪里six-vertex的传递矩阵模型在一个有限的平方晶格周期性边界条件的权重(a = b = 1)和(c > 0)。我们还表明,相同的向量(psi)满足(Hψ= E psi),在(H)的哈密顿XXZ模型(该模型是拟设的第一次开发),与一个值(E)显式计算。这种方法的变体已经成为物理学和数学界研究精确可解统计力学模型的核心技术。我们在本文中的目的是提供一个教学思想的阐述这种结构，针对数学观众。它还提供了一个机会来介绍将在作者[5]的后续论文中使用的符号和框架，这相当于证明(mathbb{Z}^{2})上具有簇权(q >4)的随机聚类模型呈现一阶相变。
{"title":"The Bethe ansatz for the six-vertex and XXZ models: An exposition","authors":"H. Duminil-Copin, Maxime Gagnebin, Matan Harel, I. Manolescu, V. Tassion","doi":"10.1214/17-PS292","DOIUrl":"https://doi.org/10.1214/17-PS292","url":null,"abstract":"In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector (psi) and energy (Lambda), which satisfy (V psi = Lambda psi), where (V) is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights (a= b=1) and (c > 0). We also show that the same vector (psi) satisfies (H psi = E psi), where (H) is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value (E) computed explicitly. \u0000 \u0000Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a \u0000pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on (mathbb{Z}^{2}) with cluster weight (q >4) exhibits a first-order phase transition. \u0000 \u0000","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"15 1","pages":"102-130"},"PeriodicalIF":1.6,"publicationDate":"2016-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-PS292","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66071434","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}
The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed (mathsf{mod}~1) sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original (p)-multiply equidistributed sequence (k^p, t {mathsf{mod}}~1), (kgeq1) (where (pinmathbb{N}) and (tin[0,1])), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations. The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to (infty)-distributed Koksma's numbers (t^k {mathsf{mod}}~1), (kgeq1) (where (tin[1,a]) for some (a>1)), and an important generalization by Niederreiter and Tichy from 1985 are discussed. The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as certain computer scientists and number theorists.
平均分配理论大约有一百年的历史，主要是由数论学家和理论计算机科学家发展起来的。由于不同学派使用的各种同义词和多义词，一个有动机的没有经验的同伴在阅读文献时可能会遇到困难。本文的目的之一是为概率学家提供一个简短的介绍。我们通过回顾2002年第二位作者的作品中的一个视角来进行。使用它，可以很容易地展示在“度量”(意味着几乎肯定是随机的)意义上的完全均匀分布(mathsf{mod}~1)序列的各种新示例。特别地，我们指出了原(p)乘等分布序列(k^p, t {mathsf{mod}}~1)， (kgeq1)(其中(pinmathbb{N})和(tin[0,1]))的自然推广，这是Hermann Weyl在1916年提出的。顺便说一下，我们还为弱完全等分布(也称为WCUD)序列导出了一个类似weyl的准则，这是最近在MCMC模拟中非常感兴趣的。从数论到概率论的转换使弱相关复值随机变量的强大数定律的一个版本成为焦点，该研究由Weyl在上述手稿中发起，由Davenport, Erdős和LeVeque在1963年跟进，并由Russell Lyons在1988年大大扩展。在此背景下，讨论了(inty)-分布Koksma数(t^k {mathsf{mod}}~1)， (kgeq1)(其中(tin[1,a])对于某些(a>1))的应用，以及Niederreiter和Tichy自1985年以来的一个重要推广。这篇论文包含的严格意义上的新数学几乎可以忽略不计，但它的观点和最后包含的开放问题可能会引起概率论家和统计学家，以及某些计算机科学家和数论学家的极大兴趣。
{"title":"Equidistribution, uniform distribution: a probabilist's perspective","authors":"V. Limic, Nedvzad Limi'c","doi":"10.1214/17-PS295","DOIUrl":"https://doi.org/10.1214/17-PS295","url":null,"abstract":"The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed (mathsf{mod}~1) sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original (p)-multiply equidistributed sequence (k^p, t {mathsf{mod}}~1), (kgeq1) (where (pinmathbb{N}) and (tin[0,1])), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations. \u0000 \u0000The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to (infty)-distributed Koksma's numbers (t^k {mathsf{mod}}~1), (kgeq1) (where (tin[1,a]) for some (a>1)), and an important generalization by Niederreiter and Tichy from 1985 are discussed. \u0000 \u0000The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as \u0000certain computer scientists and number theorists. \u0000 \u0000","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"15 1","pages":"131-155"},"PeriodicalIF":1.6,"publicationDate":"2016-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66071523","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 discrete bridge from ((0,0)) to ((2N,0)) evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order (N^{-alpha}) with (alphain(0,infty)). We provide a classification of the asymptotic behaviours - invariant measure, hydrodynamic limit and fluctuations - of this model according to the value of the parameter (alpha).
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