We prove a central limit theorem under diffusive scaling for the displacement of a random walk on ZdZd in stationary and ergodic doubly stochastic random environment, under the H−1H−1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in Lmax{2+δ,d}Lmax{2+δ,d}, with δ>0δ>0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschlager [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].
{"title":"Central limit theorem for random walks in doubly stochastic random environment: ${mathscr{H}_{-1}}$ suffices","authors":"G. Kozma, B. T'oth","doi":"10.1214/16-AOP1166","DOIUrl":"https://doi.org/10.1214/16-AOP1166","url":null,"abstract":"We prove a central limit theorem under diffusive scaling for the displacement of a random walk on ZdZd in stationary and ergodic doubly stochastic random environment, under the H−1H−1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in Lmax{2+δ,d}Lmax{2+δ,d}, with δ>0δ>0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschlager [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"4307-4347"},"PeriodicalIF":2.3,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43380177","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 present a probabilistic study of the Hilbert operator Tf(x)=1π∫∞0f(y)dyx+y,x≥0, Tf(x)=1π∫0∞f(y)dyx+y,x≥0, defined on integrable functions ff on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and ΦΦ-inequalities for TT. We also show similar estimates for higher dimensional analogues of the Hilbert operator, and by the further careful modification of martingale methods, we obtain related sharp localized inequalities for Hilbert and Riesz transforms.
{"title":"Inequalities for Hilbert operator and its extensions: The probabilistic approach","authors":"A. Osȩkowski","doi":"10.1214/15-AOP1026","DOIUrl":"https://doi.org/10.1214/15-AOP1026","url":null,"abstract":"We present a probabilistic study of the Hilbert operator Tf(x)=1π∫∞0f(y)dyx+y,x≥0, Tf(x)=1π∫0∞f(y)dyx+y,x≥0, defined on integrable functions ff on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and ΦΦ-inequalities for TT. We also show similar estimates for higher dimensional analogues of the Hilbert operator, and by the further careful modification of martingale methods, we obtain related sharp localized inequalities for Hilbert and Riesz transforms.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"535-563"},"PeriodicalIF":2.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031354","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 give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the rr-point functions for r=2,…,5r=2,…,5. The rr-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the rr-point functions, but the lack of tightness has been an obstruction to proving convergence on path space. We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.
{"title":"A criterion for convergence to super-Brownian motion on path space","authors":"R. Hofstad, Mark Holmes, E. Perkins","doi":"10.1214/14-aop953","DOIUrl":"https://doi.org/10.1214/14-aop953","url":null,"abstract":"We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the rr-point functions for r=2,…,5r=2,…,5. The rr-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the rr-point functions, but the lack of tightness has been an obstruction to proving convergence on path space. We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"278-376"},"PeriodicalIF":2.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-aop953","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66008987","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}
The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.
{"title":"Multivariate approximation in total variation, II: Discrete normal approximation","authors":"A. Barbour, M. Luczak, A. Xia","doi":"10.1214/17-AOP1205","DOIUrl":"https://doi.org/10.1214/17-AOP1205","url":null,"abstract":"The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"1405-1440"},"PeriodicalIF":2.3,"publicationDate":"2016-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1205","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66061053","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}
The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0,1]C[0,1]. These limits cannot be strengthened to convergence in L2(Ω)L2(Ω).
{"title":"Behavior of the generalized Rosenblatt process at extreme critical exponent values","authors":"Shuyang Bai, M. Taqqu","doi":"10.1214/15-AOP1087","DOIUrl":"https://doi.org/10.1214/15-AOP1087","url":null,"abstract":"The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0,1]C[0,1]. These limits cannot be strengthened to convergence in L2(Ω)L2(Ω).","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"1278-1324"},"PeriodicalIF":2.3,"publicationDate":"2016-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1087","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033600","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 the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field $X:Omegatimesmathbb{R}^dtomathbb{R}$. Let us fix a level $uin R$ and let us consider the excursion set above $u$, $A(T,u)={tin T:,X(t)ge u}$ where $T$ is a bounded cube $subset R^d$. The aim of this paper is to establish a central limit theorem for the Euler characteristic of $A(T,u)$ as $T$ grows to $R^d$, as conjectured by R. Adler more than ten years ago. The required assumption on $X$ is $C^3$ regularity of the trajectories, non degeneracy of the Gaussian vector $X(t)$ and derivatives at any fixed point $tin R^d$ as well as integrability on $R^d$ of the covariance function and its derivatives. The fact that $X$ is $C^3$ is stronger than Geman's assumption traditionally used in dimension one. Nevertheless, our result extends what is known in dimension one to higher dimension. In that case, the Euler characteristic of $A(T,u)$ equals the number of up-crossings of $X$ at level $u$, plus eventually one if $X$ is above $u$ at the left bound of the interval $T$.
{"title":"A central limit theorem for the Euler characteristic of a Gaussian excursion set","authors":"A. Estrade, J. León","doi":"10.1214/15-AOP1062","DOIUrl":"https://doi.org/10.1214/15-AOP1062","url":null,"abstract":"We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field $X:Omegatimesmathbb{R}^dtomathbb{R}$. Let us fix a level $uin R$ and let us consider the excursion set above $u$, $A(T,u)={tin T:,X(t)ge u}$ where $T$ is a bounded cube $subset R^d$. The aim of this paper is to establish a central limit theorem for the Euler characteristic of $A(T,u)$ as $T$ grows to $R^d$, as conjectured by R. Adler more than ten years ago. \u0000 \u0000The required assumption on $X$ is $C^3$ regularity of the trajectories, non degeneracy of the Gaussian vector $X(t)$ and derivatives at any fixed point $tin R^d$ as well as integrability on $R^d$ of the covariance function and its derivatives. The fact that $X$ is $C^3$ is stronger than Geman's assumption traditionally used in dimension one. Nevertheless, our result extends what is known in dimension one to higher dimension. In that case, the Euler characteristic of $A(T,u)$ equals the number of up-crossings of $X$ at level $u$, plus eventually one if $X$ is above $u$ at the left bound of the interval $T$.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"22 2 1","pages":"3849-3878"},"PeriodicalIF":2.3,"publicationDate":"2016-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1062","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033166","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}
The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multi scale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for controlling chaining functionals. The key innovation is a simple but powerful contraction principle that makes it possible to efficiently exploit interpolation. We illustrate the utility of this approach by developing new dimension-free bounds on the norms of random matrices and on chaining functionals in Banach lattices. As another application, we give a remarkably short interpolation proof of the majorizing measure theorem that entirely avoids the greedy construction that lies at the heart of earlier proofs.
{"title":"Chaining, interpolation and convexity II: The contraction principle","authors":"R. Handel","doi":"10.1214/17-AOP1214","DOIUrl":"https://doi.org/10.1214/17-AOP1214","url":null,"abstract":"The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multi scale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for controlling chaining functionals. The key innovation is a simple but powerful contraction principle that makes it possible to efficiently exploit interpolation. We illustrate the utility of this approach by developing new dimension-free bounds on the norms of random matrices and on chaining functionals in Banach lattices. As another application, we give a remarkably short interpolation proof of the majorizing measure theorem that entirely avoids the greedy construction that lies at the heart of earlier proofs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"1764-1805"},"PeriodicalIF":2.3,"publicationDate":"2016-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1214","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66061245","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 prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in the Banach space literature, including those by Johnson and Schechtman Ann. Probab. 17 (1989) 789-808], Klass and Nowicki Ann. Probab. 28 (2000) 851-862], and Hitczenko and Montgomery-Smith Ann. Probab. 29 (2001) 447-466]. Finally, we show that the Hoffmann-Jorgensen inequality (including our generalized version) holds not only in Banach spaces but more generally, in a very primitive mathematical framework required to state the inequality: a metric semigroup G. This includes normed linear spaces as well as all compact, discrete or (connected) abelian Lie groups.
{"title":"THE HOFFMANN-JORGENSEN INEQUALITY IN METRIC SEMIGROUPS","authors":"A. Khare, B. Rajaratnam","doi":"10.1214/16-AOP1160","DOIUrl":"https://doi.org/10.1214/16-AOP1160","url":null,"abstract":"We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in the Banach space literature, including those by Johnson and Schechtman Ann. Probab. 17 (1989) 789-808], Klass and Nowicki Ann. Probab. 28 (2000) 851-862], and Hitczenko and Montgomery-Smith Ann. Probab. 29 (2001) 447-466]. Finally, we show that the Hoffmann-Jorgensen inequality (including our generalized version) holds not only in Banach spaces but more generally, in a very primitive mathematical framework required to state the inequality: a metric semigroup G. This includes normed linear spaces as well as all compact, discrete or (connected) abelian Lie groups.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"4101-4111"},"PeriodicalIF":2.3,"publicationDate":"2016-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047914","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}
Let {Xk}k≥Z be a stationary sequence. Given p∈(2,3] moments and a mild weak dependence condition, we show a Berry–Esseen theorem with optimal rate np/2−1. For p≥4, we also show a convergence rate of n1/2 in Lq-norm, where q≥1. Up to logn factors, we also obtain nonuniform rates for any p>2. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics.
设{Xk}k≥Z为平稳序列。在给定p∈(2,3]矩和弱依赖条件下，我们给出了最优速率np/2−1的Berry-Esseen定理。当p≥4时，我们也证明了在q≥1的lq范数上的收敛速率为n1/2。对于任意p / b / 2，我们也得到了非均匀速率。这导致了从时间序列文献中许多线性和非线性过程的新的最优结果，但也包括动力系统理论的例子。证明是基于特征函数、耦合和条件参数和理想度量的混合方法。
{"title":"Berry–Esseen theorems under weak dependence","authors":"M. Jirak","doi":"10.1214/15-AOP1017","DOIUrl":"https://doi.org/10.1214/15-AOP1017","url":null,"abstract":"Let {Xk}k≥Z be a stationary sequence. Given p∈(2,3] moments and a mild weak dependence condition, we show a Berry–Esseen theorem with optimal rate np/2−1. For p≥4, we also show a convergence rate of n1/2 in Lq-norm, where q≥1. Up to logn factors, we also obtain nonuniform rates for any p>2. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"2024-2063"},"PeriodicalIF":2.3,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031755","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":"Imaginary geometry II: Reversibility of $operatorname{SLE}_{kappa}(rho_{1};rho_{2})$ for $kappain(0,4)$","authors":"Jason Miller, S. Sheffield","doi":"10.1214/14-AOP943","DOIUrl":"https://doi.org/10.1214/14-AOP943","url":null,"abstract":"","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"1647-1722"},"PeriodicalIF":2.3,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP943","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66007930","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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