We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition (T ) of Sznitman (cf. [Sz01]). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni [CZ02]. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.
{"title":"On the transient (T) condition for random walk in mixing environment","authors":"E. Aguilar","doi":"10.1214/18-AOP1330","DOIUrl":"https://doi.org/10.1214/18-AOP1330","url":null,"abstract":"We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition (T ) of Sznitman (cf. [Sz01]). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni [CZ02]. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77094245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For d≥2d≥2 and n∈Nn∈N even, let pn=pn(d)pn=pn(d) denote the number of length nn self-avoiding polygons in ZdZd up to translation. The polygon cardinality grows exponentially, and the growth rate limn∈2Np1/nn∈(0,∞)limn∈2Npn1/n∈(0,∞) is called the connective constant and denoted by μμ. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that pnμ−n≤Cn−1/2pnμ−n≤Cn−1/2 in dimension d=2d=2. Here, we establish that pnμ−n≤n−3/2+o(1)pnμ−n≤n−3/2+o(1) for a set of even nn of full density when d=2d=2. We also consider a certain variant of self-avoiding walk and argue that, when d≥3d≥3, an upper bound of n−2+d−1+o(1)n−2+d−1+o(1) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.
{"title":"An upper bound on the number of self-avoiding polygons via joining","authors":"A. Hammond","doi":"10.1214/17-AOP1182","DOIUrl":"https://doi.org/10.1214/17-AOP1182","url":null,"abstract":"For d≥2d≥2 and n∈Nn∈N even, let pn=pn(d)pn=pn(d) denote the number of length nn self-avoiding polygons in ZdZd up to translation. The polygon cardinality grows exponentially, and the growth rate limn∈2Np1/nn∈(0,∞)limn∈2Npn1/n∈(0,∞) is called the connective constant and denoted by μμ. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that pnμ−n≤Cn−1/2pnμ−n≤Cn−1/2 in dimension d=2d=2. Here, we establish that pnμ−n≤n−3/2+o(1)pnμ−n≤n−3/2+o(1) for a set of even nn of full density when d=2d=2. We also consider a certain variant of self-avoiding walk and argue that, when d≥3d≥3, an upper bound of n−2+d−1+o(1)n−2+d−1+o(1) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2018-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1182","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48814709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous and which we therefore refer to as Aldous diffusion, as a solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.
{"title":"The Aldous chain on cladograms in the diffusion limit","authors":"Wolfgang Lohr, L. Mytnik, A. Winter","doi":"10.1214/20-AOP1431","DOIUrl":"https://doi.org/10.1214/20-AOP1431","url":null,"abstract":"In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous and which we therefore refer to as Aldous diffusion, as a solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2018-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46622993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Benson Au, Guillaume C'ebron, Antoine Dahlqvist, Franck Gabriel, C. Male
We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erdős-Renyi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).
{"title":"Freeness over the diagonal for large random matrices","authors":"Benson Au, Guillaume C'ebron, Antoine Dahlqvist, Franck Gabriel, C. Male","doi":"10.1214/20-AOP1447","DOIUrl":"https://doi.org/10.1214/20-AOP1447","url":null,"abstract":"We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erdős-Renyi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2018-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66080625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erdős–Renyi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the “spectral redemption conjecture” in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.
{"title":"Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs","authors":"C. Bordenave, M. Lelarge, L. Massoulié","doi":"10.1214/16-AOP1142","DOIUrl":"https://doi.org/10.1214/16-AOP1142","url":null,"abstract":"A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erdős–Renyi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the “spectral redemption conjecture” in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1142","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.
{"title":"Existence conditions of permanental and multivariate negative binomial distributions","authors":"Nathalie Eisenbaum, F. Maunoury","doi":"10.1214/17-AOP1179","DOIUrl":"https://doi.org/10.1214/17-AOP1179","url":null,"abstract":"Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1179","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42298657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present some new limit theorems for power variation of kkth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k≥1k≥1, the considered power p>0p>0, the Blumenthal–Getoor index β∈[0,2)β∈[0,2) of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process LL is a symmetric ββ-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k−α)β(k−α)β-stable totally right skewed random variable.
{"title":"Power variation for a class of stationary increments Lévy driven moving averages","authors":"A. Basse-O’Connor, R. Lachièze-Rey, M. Podolskij","doi":"10.1214/16-AOP1170","DOIUrl":"https://doi.org/10.1214/16-AOP1170","url":null,"abstract":"In this paper, we present some new limit theorems for power variation of kkth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k≥1k≥1, the considered power p>0p>0, the Blumenthal–Getoor index β∈[0,2)β∈[0,2) of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process LL is a symmetric ββ-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k−α)β(k−α)β-stable totally right skewed random variable.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1170","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45244732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles’ masses decay. In standard BBM, we may define the front displacement at time tt as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from Θ(1)Θ(1) to o(1)o(1). We show that one can find arbitrarily large times tt for which this occurs at a distance Θ(t1/3)Θ(t1/3) behind the front displacement for standard BBM.
{"title":"The front location in branching Brownian motion with decay of mass","authors":"L. Addario-Berry, S. Penington","doi":"10.1214/16-AOP1148","DOIUrl":"https://doi.org/10.1214/16-AOP1148","url":null,"abstract":"We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles’ masses decay. In standard BBM, we may define the front displacement at time tt as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from Θ(1)Θ(1) to o(1)o(1). We show that one can find arbitrarily large times tt for which this occurs at a distance Θ(t1/3)Θ(t1/3) behind the front displacement for standard BBM.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1148","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46509277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a Clark–Ocone formula for square-integrable functionals of a general temporal point process satisfying only a mild moment condition, generalizing known results on the Poisson space. Some classical applications are given, namely a deviation bound and the construction of a hedging portfolio in a pure-jump market model. As a more modern application, we provide a bound on the total variation distance between two temporal point processes, improving in some sense a recent result in this direction.
{"title":"A Clark–Ocone formula for temporal point processes and applications","authors":"I. Flint, G. Torrisi","doi":"10.1214/16-AOP1136","DOIUrl":"https://doi.org/10.1214/16-AOP1136","url":null,"abstract":"We provide a Clark–Ocone formula for square-integrable functionals of a general temporal point process satisfying only a mild moment condition, generalizing known results on the Poisson space. Some classical applications are given, namely a deviation bound and the construction of a hedging portfolio in a pure-jump market model. As a more modern application, we provide a bound on the total variation distance between two temporal point processes, improving in some sense a recent result in this direction.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1136","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47283691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568–608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.
本文研究了种群遗传学中出现的一类退化扩散算子的各种概率和解析性质，即广义Kimura扩散算子Epstein和Mazzeo [SIAM J. Math]。肛门42 (2010)568-608;种群生物学中的退化扩散算子(2013)普林斯顿大学出版社;应用数学研究快报(2016)。我们的主要成果是退化低阶系数奇异抛物方程弱解的随机表示和Kimura抛物方程非负解的尺度不变Harnack不等式的证明。我们所建立的解的随机表示是对经典费曼-卡茨公式关于扩散矩阵的简并性、漂移系数的有界性和弱解的先验正则性等假设的结果的一个相当大的推广。
{"title":"The Feynman–Kac formula and Harnack inequality for degenerate diffusions","authors":"C. Epstein, C. Pop","doi":"10.1214/16-AOP1138","DOIUrl":"https://doi.org/10.1214/16-AOP1138","url":null,"abstract":"We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568–608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1138","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46532803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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