{"title":"A lower bound for disconnection by simple random walk","authors":"Xinyi Li","doi":"10.1214/15-AOP1077","DOIUrl":"https://doi.org/10.1214/15-AOP1077","url":null,"abstract":"We consider simple random walk on ZdZd, d≥3d≥3. Motivated by the work of A.-S. Sznitman and the author in [Probab. Theory Related Fields 161 (2015) 309–350] and [Electron. J. Probab. 19 (2014) 1–26], we investigate the asymptotic behavior of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk. We derive asymptotic lower bounds that bring into play random interlacements. Although open at the moment, some of the lower bounds we obtain possibly match the asymptotic upper bounds recently obtained in [Disconnection, random walks, and random interlacements (2014)]. This potentially yields special significance to the tilted walks that we use in this work, and to the strategy that we employ to implement disconnection.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1077","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032936","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}
{"title":"Central limit theorems and bootstrap in high dimensions","authors":"V. Chernozhukov, D. Chetverikov, Kengo Kato","doi":"10.1920/WP.CEM.2016.3916","DOIUrl":"https://doi.org/10.1920/WP.CEM.2016.3916","url":null,"abstract":"In this paper, we derive central limit and bootstrap theorems for probabilities that centered high-dimensional vector sums hit rectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for the probabilities that a root-n rescaled sample average of Xi is in A, where X1,..., Xn are independent random vectors in Rp and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn-> infinity and p>>n; in particular, p can be as large as O(e^(Cn^c)) for some constants c,C>0. The result holds uniformly over all rectangles, or more generally, sparsely convex sets, and does not require any restrictions on the correlation among components of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend nontrivially only on a small subset of their arguments, with rectangles being a special case.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68013037","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}
D. Buraczewski, Jeffrey F. Collamore, E. Damek, J. Zienkiewicz
{"title":"Large deviation estimates for exceedance times of perpetuity sequences and their dual processes","authors":"D. Buraczewski, Jeffrey F. Collamore, E. Damek, J. Zienkiewicz","doi":"10.1214/15-AOP1059","DOIUrl":"https://doi.org/10.1214/15-AOP1059","url":null,"abstract":"In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence Yn:=B1+A1B2+⋯+(A1⋯An−1)BnYn:=B1+A1B2+⋯+(A1⋯An−1)Bn, where (Ai,Bi)⊂(0,∞)×R(Ai,Bi)⊂(0,∞)×R. Estimates for the stationary tail distribution of {Yn}{Yn} have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207–248] and Goldie [Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that if M:=supnYnM:=supnYn, then P{M>u}∼CMu−ξP{M>u}∼CMu−ξ as u→∞u→∞. While much attention has been focused on extending such estimates to more general settings, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the normalized first passage time Tu:=(logu)−1inf{n:Yn>u}Tu:=(logu)−1inf{n:Yn>u}. We begin by showing that, conditional on {Tu<∞}{Tu<∞}, Tu→ρTu→ρ as u→∞u→∞ for a certain positive constant ρρ. We then provide a conditional central limit theorem for {Tu}{Tu}, and study P{Tu∈G}P{Tu∈G} as u→∞u→∞ for sets G⊂[0,∞)G⊂[0,∞). If G⊂[0,ρ)G⊂[0,ρ), then we show that P{Tu∈G}uI(G)→C(G)P{Tu∈G}uI(G)→C(G) as u→∞u→∞ for a certain large deviation rate function II and constant C(G)C(G). On the other hand, if G⊂(ρ,∞)G⊂(ρ,∞), then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac [In Random Matrices and Their Applications (Brunswick, Maine, 1984) (1986) 263–273 Amer. Math. Soc.], namely to the reflected process M∗n:=max{AnM∗n−1+Bn,0}Mn∗:=max{AnMn−1∗+Bn,0}, n∈Z+n∈Z+. Using Siegmund duality, we relate the first passage times of {Yn}{Yn} to the finite-time exceedance probabilities of {M∗n}{Mn∗}, yielding a new result concerning the convergence of {M∗n}{Mn∗} to its stationary distribution.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032696","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}
{"title":"Dissipation and high disorder","authors":"Le Chen, M. Cranston, D. Khoshnevisan, Kunwoo Kim","doi":"10.1214/15-AOP1040","DOIUrl":"https://doi.org/10.1214/15-AOP1040","url":null,"abstract":"Given a fieldfB(x)gx2Zd of independent standard Brownian motions, indexed by Z d , the generator of a suitable Markov process on Z d ; G; and suciently nice function : [0;1)! [0;1); we consider the influence of the parameter on the behavior of the system, dut(x) = (Gut)(x) dt + (ut(x))dBt(x) [t > 0; x2 Z d ]; u0(x) = c0 0(x); We show that for any","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031932","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}
{"title":"Tail estimates for Markovian rough paths","authors":"T. Cass, M. Ogrodnik","doi":"10.1214/16-AOP1117","DOIUrl":"https://doi.org/10.1214/16-AOP1117","url":null,"abstract":"We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms ([26]) and prove a better-than-exponential tail estimate for the accumulated local p-variation functional, which has been introduced and studied in [17]. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao [32], and Chevyrev and Lyons [18].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1117","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047729","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}
{"title":"BSE’s, BSDE’s and fixed-point problems","authors":"Patrick Cheridito, Kihun Nam","doi":"10.1214/16-AOP1149","DOIUrl":"https://doi.org/10.1214/16-AOP1149","url":null,"abstract":"In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed-point problem in a space of random vectors. This makes it possible to employ general fixed-point arguments to establish the existence of a solution. For instance, Banach’s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed-point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean–Vlasov-type equations and equations with coefficients of superlinear growth.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1149","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047820","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}
{"title":"A quantitative Burton-Keane estimate under strong FKG condition","authors":"H. Duminil-Copin, D. Ioffe, Y. Velenik","doi":"10.1214/15-AOP1049","DOIUrl":"https://doi.org/10.1214/15-AOP1049","url":null,"abstract":"We consider translationally-invariant percolation models on ZdZd satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance nn (this corresponds to a finite size version of the celebrated Burton–Keane [Comm. Math. Phys. 121 (1989) 501–505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincare inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight q≥1q≥1.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032735","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}
{"title":"The cut-and-paste process","authors":"Harry Crane","doi":"10.1214/14-AOP922","DOIUrl":"https://doi.org/10.1214/14-AOP922","url":null,"abstract":"We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a $sigma$-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a L'evy-It^o representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP922","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66006877","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}
{"title":"Efron–Stein inequalities for random matrices","authors":"D. Paulin, Lester W. Mackey, J. Tropp","doi":"10.1214/15-AOP1054","DOIUrl":"https://doi.org/10.1214/15-AOP1054","url":null,"abstract":"This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron–Stein inequalities developed by Boucheron et al. The proofs rely on the method of exchangeable pairs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032488","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}
Elisabetta Candellero, S. Ganguly, C. Hoffman, Lionel Levine
{"title":"Oil and water: A two-type internal aggregation model","authors":"Elisabetta Candellero, S. Ganguly, C. Hoffman, Lionel Levine","doi":"10.1214/16-AOP1157","DOIUrl":"https://doi.org/10.1214/16-AOP1157","url":null,"abstract":"We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting withn oil\" andn water\" particles at the origin, the particles diuse in Z according to the following rule: whenever some site x2 Z has at least 1 oil and at least 1 water particle present, it res by sending 1 oil particle and 1 water particle each to an independent random neighbor x 1. Firing continues until every site has at most one type of particles. We establish the correct order for several statistics of this model and identify the scaling limit under assumption of existence.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2014-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047864","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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