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Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
http://projecteuclid.org/euclid.aop/1278593952
<strong>Terrence M. Adams</strong>, <strong>Andrew B. Nobel</strong><p><strong>Source: </strong>Ann. Probab., Volume 38, Number 4, 1345--1367.</p><p><strong>Abstract:</strong><br/>
We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$ , the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$ . The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.
</p>projecteuclid.org/euclid.aop/1278593952_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA leader-election procedure using recordshttps://projecteuclid.org/euclid.aop/1513069262<strong>Gerold Alsmeyer</strong>, <strong>Zakhar Kabluchko</strong>, <strong>Alexander Marynych</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4348--4388.</p><p><strong>Abstract:</strong><br/>
Motivated by the open problem of finding the asymptotic distributional behavior of the number of collisions in a Poisson–Dirichlet coalescent, the following version of a stochastic leader-election algorithm is studied. Consider an infinite family of persons, labeled by $1,2,3,\ldots$, who generate i.i.d. random numbers from an arbitrary continuous distribution. Those persons who have generated a record value, that is, a value larger than the values of all previous persons, stay in the game, all others must leave. The remaining persons are relabeled by $1,2,3,\ldots$ maintaining their order in the first round, and the election procedure is repeated independently from the past and indefinitely. We prove limit theorems for a number of relevant functionals for this procedure, notably the number of rounds $T(M)$ until all persons among $1,\ldots,M$, except the first one, have left (as $M\to\infty$). For example, we show that the sequence $(T(M)-\log^{*}M)_{M\in \mathbb{N}}$, where $\log^{*}$ denotes the iterated logarithm, is tight, and study its weak subsequential limits. We further provide an appropriate and apparently new kind of normalization (based on tetrations) such that the original labels of persons who stay in the game until round $n$ converge (as $n\to\infty$) to some random non-Poissonian point process and study its properties. The results are applied to describe all subsequential distributional limits for the number of collisions in the Poisson–Dirichlet coalescent, thus providing a complete answer to the open problem mentioned above.
</p>projecteuclid.org/euclid.aop/1513069262_20171212220040Tue, 12 Dec 2017 22:00 ESTLévy processes and Lévy white noise as tempered distributionshttps://projecteuclid.org/euclid.aop/1513069263<strong>Robert C. Dalang</strong>, <strong>Thomas Humeau</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4389--4418.</p><p><strong>Abstract:</strong><br/>
We identify a necessary and sufficient condition for a Lévy white noise to be a tempered distribution. More precisely, we show that if the Lévy measure associated with this noise has a positive absolute moment, then the Lévy white noise almost surely takes values in the space of tempered distributions. If the Lévy measure does not have a positive absolute moment of any order, then the event on which the Lévy white noise is a tempered distribution has probability zero.
</p>projecteuclid.org/euclid.aop/1513069263_20171212220040Tue, 12 Dec 2017 22:00 ESTLarge deviations for random projections of $\ell^{p}$ ballshttps://projecteuclid.org/euclid.aop/1513069264<strong>Nina Gantert</strong>, <strong>Steven Soojin Kim</strong>, <strong>Kavita Ramanan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4419--4476.</p><p><strong>Abstract:</strong><br/>
Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^{p}$ ball in $\mathbb{R}^{n}$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$), the corresponding rate function is “universal,” in the sense that it coincides for “almost every” sequence of projection directions. We also analyze some exceptional sequences of directions in the “measure zero” set, including the sequence of directions corresponding to the classical Cramér’s theorem, and show that those sequences of directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of $\ell^{p}$ balls.
</p>projecteuclid.org/euclid.aop/1513069264_20171212220040Tue, 12 Dec 2017 22:00 ESTPower variation for a class of stationary increments Lévy driven moving averageshttps://projecteuclid.org/euclid.aop/1513069265<strong>Andreas Basse-O’Connor</strong>, <strong>Raphaël Lachièze-Rey</strong>, <strong>Mark Podolskij</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4477--4528.</p><p><strong>Abstract:</strong><br/>
In this paper, we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy 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 $k\geq1$, the considered power $p>0$, the Blumenthal–Getoor index $\beta\in[0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $\alpha$. 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 Lévy process $L$ is a symmetric $\beta$-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a $(k-\alpha)\beta$-stable totally right skewed random variable.
</p>projecteuclid.org/euclid.aop/1513069265_20171212220040Tue, 12 Dec 2017 22:00 ESTEquilibrium fluctuation of the Atlas modelhttps://projecteuclid.org/euclid.aop/1513069266<strong>Amir Dembo</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4529--4560.</p><p><strong>Abstract:</strong><br/>
We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($\mathbb{Z}_{+}$-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $\mathbb{R}_{+}$. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-\frac{1}{4}}$, converges as $t\to\infty$ to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on $\mathbb{R}_{+}$ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [ Ann. Appl. Probab. 18 (2008) 2179–2207] about the asymptotic Gaussian fluctuation of the ranked particles.
</p>projecteuclid.org/euclid.aop/1513069266_20171212220040Tue, 12 Dec 2017 22:00 ESTStochastic heat equation with rough dependence in spacehttps://projecteuclid.org/euclid.aop/1513069267<strong>Yaozhong Hu</strong>, <strong>Jingyu Huang</strong>, <strong>Khoa Lê</strong>, <strong>David Nualart</strong>, <strong>Samy Tindel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4561--4616.</p><p><strong>Abstract:</strong><br/>
This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable. The existence and uniqueness of the solution $u$ are proved assuming the nonlinear coefficient $\sigma(u)$ is differentiable with a Lipschitz derivative and $\sigma(0)=0$.
</p>projecteuclid.org/euclid.aop/1513069267_20171212220040Tue, 12 Dec 2017 22:00 ESTParisi formula for the ground state energy in the mixed $p$-spin modelhttps://projecteuclid.org/euclid.aop/1513069268<strong>Antonio Auffinger</strong>, <strong>Wei-Kuo Chen</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4617--4631.</p><p><strong>Abstract:</strong><br/>
We show that the thermodynamic limit of the ground state energy in the mixed $p$-spin model can be identified as a variational problem. This gives a natural generalization of the Parisi formula at zero temperature.
</p>projecteuclid.org/euclid.aop/1513069268_20171212220040Tue, 12 Dec 2017 22:00 ESTInvariance timeshttps://projecteuclid.org/euclid.aop/1513069269<strong>Stéphane Crépey</strong>, <strong>Shiqi Song</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4632--4674.</p><p><strong>Abstract:</strong><br/>
On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$, we consider two filtrations $\mathbb{F}\subset\mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup, it is well known that, for any $\mathbb{F}$ semimartingale $X$, the process $X^{\theta-}$ ($X$ stopped “right before $\theta$”) is a $\mathbb{G}$ semimartingale. Given a positive constant $T$, we call $\theta$ an invariance time if there exists a probability measure $\mathbb{P}$ equivalent to $\mathbb{Q}$ on $\mathcal{F}_{T}$ such that, for any $(\mathbb{F},\mathbb{P})$ local martingale $X$, $X^{\theta-}$ is a $(\mathbb{G},\mathbb{Q})$ local martingale. We characterize invariance times in terms of the $(\mathbb{F},\mathbb{Q})$ Azéma supermartingale of $\theta$ and we derive a mild and tractable invariance time sufficiency condition. We discuss invariance times in mathematical finance and BSDE applications.
</p>projecteuclid.org/euclid.aop/1513069269_20171212220040Tue, 12 Dec 2017 22:00 ESTAsymptotic expansion of the invariant measure for ballistic random walk in the low disorder regimehttps://projecteuclid.org/euclid.aop/1513069270<strong>David Campos</strong>, <strong>Alejandro F. Ramírez</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4675--4699.</p><p><strong>Abstract:</strong><br/>
We consider a random walk in random environment in the low disorder regime on $\mathbb{Z}^{d}$, that is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\varepsilon\xi(x,e)$, where $p(e)$ is deterministic, $\{\{\xi(x,e):\vert e\vert_{1}=1\}:x\in\mathbb{Z}^{d}\}$ are i.i.d. and $\varepsilon>0$ is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in $\varepsilon$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in $\varepsilon$ for the invariant measure of random perturbations of the simple symmetric random walk in dimensions $d=2$.
</p>projecteuclid.org/euclid.aop/1513069270_20171212220040Tue, 12 Dec 2017 22:00 ESTPolarity of points for Gaussian random fieldshttps://projecteuclid.org/euclid.aop/1513069271<strong>Robert C. Dalang</strong>, <strong>Carl Mueller</strong>, <strong>Yimin Xiao</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4700--4751.</p><p><strong>Abstract:</strong><br/>
We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions $k\geq1$. Our approach builds on a delicate covering argument developed by M. Talagrand [ Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.
</p>projecteuclid.org/euclid.aop/1513069271_20171212220040Tue, 12 Dec 2017 22:00 ESTThe vacant set of two-dimensional critical random interlacement is infinitehttps://projecteuclid.org/euclid.aop/1513069272<strong>Francis Comets</strong>, <strong>Serguei Popov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4752--4785.</p><p><strong>Abstract:</strong><br/>
For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s. infinite, thus solving an open problem from [ Commun. Math. Phys. 343 (2016) 129–164]. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.
</p>projecteuclid.org/euclid.aop/1513069272_20171212220040Tue, 12 Dec 2017 22:00 ESTExistence conditions of permanental and multivariate negative binomial distributionshttps://projecteuclid.org/euclid.aop/1513069273<strong>Nathalie Eisenbaum</strong>, <strong>Franck Maunoury</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 45, Number 6b, 4786--4820.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.aop/1513069273_20171212220040Tue, 12 Dec 2017 22:00 ESTNonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphshttps://projecteuclid.org/euclid.aop/1517821218<strong>Charles Bordenave</strong>, <strong>Marc Lelarge</strong>, <strong>Laurent Massoulié</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 1--71.</p><p><strong>Abstract:</strong><br/>
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–Rényi 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.
</p>projecteuclid.org/euclid.aop/1517821218_20180205040027Mon, 05 Feb 2018 04:00 ESTSize biased couplings and the spectral gap for random regular graphshttps://projecteuclid.org/euclid.aop/1517821219<strong>Nicholas Cook</strong>, <strong>Larry Goldstein</strong>, <strong>Tobias Johnson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 72--125.</p><p><strong>Abstract:</strong><br/>
Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\lambda=2\sqrt{d-1}+o(1)$ with high probability. In the present work, we show that $\lambda=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress toward a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.
</p>projecteuclid.org/euclid.aop/1517821219_20180205040027Mon, 05 Feb 2018 04:00 ESTPath-dependent equations and viscosity solutions in infinite dimensionhttps://projecteuclid.org/euclid.aop/1517821220<strong>Andrea Cosso</strong>, <strong>Salvatore Federico</strong>, <strong>Fausto Gozzi</strong>, <strong>Mauro Rosestolato</strong>, <strong>Nizar Touzi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 126--174.</p><p><strong>Abstract:</strong><br/>
Path-dependent partial differential equations (PPDEs) are natural objects to study when one deals with non-Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus [see Dupire (2009)], in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions [see, e.g., Dupire (2009) and Cont (2016) Stochastic Integration by Parts and Functional Itô Calculus 115–207, Birkhäuser] and viscosity solutions [see, e.g., Ekren et al. (2014) Ann. Probab. 42 204–236]. In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.
</p>projecteuclid.org/euclid.aop/1517821220_20180205040027Mon, 05 Feb 2018 04:00 ESTAn upper bound on the number of self-avoiding polygons via joininghttps://projecteuclid.org/euclid.aop/1517821221<strong>Alan Hammond</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 175--206.</p><p><strong>Abstract:</strong><br/>
For $d\geq2$ and $n\in\mathbb{N}$ even, let $p_{n}=p_{n}(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^{d}$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n\in2\mathbb{N}}p_{n}^{1/n}\in(0,\infty)$ is called the connective constant and denoted by $\mu$. Madras [ J. Stat. Phys. 78 (1995) 681–699] has shown that $p_{n}\mu^{-n}\leq Cn^{-1/2}$ in dimension $d=2$. Here, we establish that $p_{n}\mu^{-n}\leq n^{-3/2+o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of self-avoiding walk and argue that, when $d\geq3$, an upper bound of $n^{-2+d^{-1}+o(1)}$ holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.
</p>projecteuclid.org/euclid.aop/1517821221_20180205040027Mon, 05 Feb 2018 04:00 ESTRandom planar maps and growth-fragmentationshttps://projecteuclid.org/euclid.aop/1517821222<strong>Jean Bertoin</strong>, <strong>Nicolas Curien</strong>, <strong>Igor Kortchemski</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 207--260.</p><p><strong>Abstract:</strong><br/>
We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.
</p>projecteuclid.org/euclid.aop/1517821222_20180205040027Mon, 05 Feb 2018 04:00 ESTDimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalitieshttps://projecteuclid.org/euclid.aop/1517821223<strong>François Bolley</strong>, <strong>Ivan Gentil</strong>, <strong>Arnaud Guillin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 261--301.</p><p><strong>Abstract:</strong><br/>
In this work, we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities. For this, we use optimal transport methods and the Borell–Brascamp–Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behavior for the laws of solutions to stochastic differential equations.
</p>projecteuclid.org/euclid.aop/1517821223_20180205040027Mon, 05 Feb 2018 04:00 ESTQuenched invariance principle for random walks with time-dependent ergodic degenerate weightshttps://projecteuclid.org/euclid.aop/1517821224<strong>Sebastian Andres</strong>, <strong>Alberto Chiarini</strong>, <strong>Jean-Dominique Deuschel</strong>, <strong>Martin Slowik</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 302--336.</p><p><strong>Abstract:</strong><br/>
We study a continuous-time random walk, $X$, on $\mathbb{Z}^{d}$ in an environment of dynamic random conductances taking values in $(0,\infty)$. We assume that the law of the conductances is ergodic with respect to space–time shifts. We prove a quenched invariance principle for the Markov process $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser’s iteration scheme.
</p>projecteuclid.org/euclid.aop/1517821224_20180205040027Mon, 05 Feb 2018 04:00 ESTAn $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergencehttps://projecteuclid.org/euclid.aop/1517821225<strong>Christian Borgs</strong>, <strong>Jennifer T. Chayes</strong>, <strong>Henry Cohn</strong>, <strong>Yufei Zhao</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 337--396.</p><p><strong>Abstract:</strong><br/>
We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.
</p>projecteuclid.org/euclid.aop/1517821225_20180205040027Mon, 05 Feb 2018 04:00 ESTLattice approximation to the dynamical $\Phi_{3}^{4}$ modelhttps://projecteuclid.org/euclid.aop/1517821226<strong>Rongchan Zhu</strong>, <strong>Xiangchan Zhu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 397--455.</p><p><strong>Abstract:</strong><br/>
We study the lattice approximations to the dynamical $\Phi^{4}_{3}$ model by paracontrolled distributions proposed in [ Forum Math. Pi 3 (2015) e6]. We prove that the solutions to the lattice systems converge to the solution to the dynamical $\Phi_{3}^{4}$ model in probability, locally uniformly in time. Since the dynamical $\Phi_{3}^{4}$ model is not well defined in the classical sense and renormalisation has to be performed in order to define the nonlinear term, a corresponding suitable drift term is added in the stochastic equations for the lattice systems.
</p>projecteuclid.org/euclid.aop/1517821226_20180205040027Mon, 05 Feb 2018 04:00 ESTRandom walks on the random graphhttps://projecteuclid.org/euclid.aop/1517821227<strong>Nathanaël Berestycki</strong>, <strong>Eyal Lubetzky</strong>, <strong>Yuval Peres</strong>, <strong>Allan Sly</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 456--490.</p><p><strong>Abstract:</strong><br/>
We study random walks on the giant component of the Erdős–Rényi random graph $\mathcal{G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $\log^{2}n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(\log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $(\nu\mathbf{d})^{-1}\log n\pm(\log n)^{1/2+o(1)}$, where $\nu$ and $\mathbf{d}$ are the speed of random walk and dimension of harmonic measure on a $\operatorname{Poisson}(\lambda)$-Galton–Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the nonbacktracking random walk.
</p>projecteuclid.org/euclid.aop/1517821227_20180205040027Mon, 05 Feb 2018 04:00 ESTA class of globally solvable Markovian quadratic BSDE systems and applicationshttps://projecteuclid.org/euclid.aop/1517821228<strong>Hao Xing</strong>, <strong>Gordan Žitković</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 491--550.</p><p><strong>Abstract:</strong><br/>
We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games and martingales on Riemannian manifolds.
</p>projecteuclid.org/euclid.aop/1517821228_20180205040027Mon, 05 Feb 2018 04:00 ESTStochastic control for a class of nonlinear kernels and applicationshttps://projecteuclid.org/euclid.aop/1517821229<strong>Dylan Possamaï</strong>, <strong>Xiaolu Tan</strong>, <strong>Chao Zhou</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 1, 551--603.</p><p><strong>Abstract:</strong><br/>
We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimization, over a set of possibly nondominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalizations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semimartingale characterization of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in Soner, Touzi and Zhang [ Probab. Theory Related Fields 153 (2012) 149–190]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of Nutz [ Stochastic Process. Appl. 125 (2015) 4543–4555], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path–dependent partial differential equation (PPDE).
</p>projecteuclid.org/euclid.aop/1517821229_20180205040027Mon, 05 Feb 2018 04:00 ESTScaling limits for sub-ballistic biased random walks in random conductanceshttps://projecteuclid.org/euclid.aop/1520586267<strong>Alexander Fribergh</strong>, <strong>Daniel Kious</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 605--686.</p><p><strong>Abstract:</strong><br/>
We consider biased random walks in positive random conductances on the $d$-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional law of large numbers for the position of the walker, properly rescaled. Moreover, we state a functional central limit theorem where an atypical process, related to the fractional kinetics, appears in the limit.
</p>projecteuclid.org/euclid.aop/1520586267_20180309040445Fri, 09 Mar 2018 04:04 ESTGrowth exponent for loop-erased random walk in three dimensionshttps://projecteuclid.org/euclid.aop/1520586268<strong>Daisuke Shiraishi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 687--774.</p><p><strong>Abstract:</strong><br/>
Let $M_{n}$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^{3}$ run until its first exit from a ball of radius $n$. In the paper, we will show the existence of the growth exponent, that is, we show that there exists $\beta>0$ such that \begin{equation*}\lim_{n\to\infty}\frac{\log E(M_{n})}{\log n}=\beta.\end{equation*}
</p>projecteuclid.org/euclid.aop/1520586268_20180309040445Fri, 09 Mar 2018 04:04 ESTLarge deviations of the trajectory of empirical distributions of Feller processes on locally compact spaceshttps://projecteuclid.org/euclid.aop/1520586269<strong>Richard C. Kraaij</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 775--828.</p><p><strong>Abstract:</strong><br/>
We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform.
We prove the large deviation principle and show that the rate function can be decomposed as a rate function for the initial time and an integral over the Lagrangian, finite only for absolutely continuous trajectories of measures.
We apply this result for diffusion and Lévy processes on $\mathbb{R}^{d}$, for pure jump processes with bounded jump kernel on arbitrary locally compact spaces and for discrete interacting particle systems. For diffusion processes, the theorem partly extends the Dawson and Gärtner theorem for noninteracting copies in the sense that it only holds for time-homogeneous processes, but on the other hand it holds for processes with degenerate diffusion matrix.
</p>projecteuclid.org/euclid.aop/1520586269_20180309040445Fri, 09 Mar 2018 04:04 ESTFree energy in the Potts spin glasshttps://projecteuclid.org/euclid.aop/1520586270<strong>Dmitry Panchenko</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 829--864.</p><p><strong>Abstract:</strong><br/>
We study the Potts spin glass model, which generalizes the Sherrington–Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product, which includes the Potts spin glass as a special case. As another example of application, one can show that Talagrand’s bound for multiple copies of the mixed $p$-spin model with constrained overlaps is asymptotically sharp. We will consider these problems in the subsequent paper and illustrate the main new idea on the technically more transparent case of the Potts spin glass.
</p>projecteuclid.org/euclid.aop/1520586270_20180309040445Fri, 09 Mar 2018 04:04 ESTFree energy in the mixed $p$-spin models with vector spinshttps://projecteuclid.org/euclid.aop/1520586271<strong>Dmitry Panchenko</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 865--896.</p><p><strong>Abstract:</strong><br/>
Using the synchronization mechanism developed in the previous work on the Potts spin glass model, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand’s upper bound for the free energy of multiple mixed $p$-spin systems coupled by constraining their overlaps.
</p>projecteuclid.org/euclid.aop/1520586271_20180309040445Fri, 09 Mar 2018 04:04 ESTA fractional kinetic process describing the intermediate time behaviour of cellular flowshttps://projecteuclid.org/euclid.aop/1520586272<strong>Martin Hairer</strong>, <strong>Gautam Iyer</strong>, <strong>Leonid Koralov</strong>, <strong>Alexei Novikov</strong>, <strong>Zsolt Pajor-Gyulai</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 897--955.</p><p><strong>Abstract:</strong><br/>
This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: a Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin–Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales.
As a consequence of our main theorem, we obtain a homogenization result for the associated advection diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.
</p>projecteuclid.org/euclid.aop/1520586272_20180309040445Fri, 09 Mar 2018 04:04 ESTQuasi-symmetries of determinantal point processeshttps://projecteuclid.org/euclid.aop/1520586273<strong>Alexander I. Bufetov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 956--1003.</p><p><strong>Abstract:</strong><br/>
The main result of this paper is that determinantal point processes on $\mathbb{R}$ corresponding to projection operators with integrable kernels are quasi-invariant, in the continuous case, under the group of diffeomorphisms with compact support (Theorem 1.4); in the discrete case, under the group of all finite permutations of the phase space (Theorem 1.6). The Radon–Nikodym derivative is computed explicitly and is given by a regularized multiplicative functional. Theorem 1.4 applies, in particular, to the sine-process, as well as to determinantal point processes with the Bessel and the Airy kernels; Theorem 1.6 to the discrete sine-process and the Gamma kernel process. The paper answers a question of Grigori Olshanski.
</p>projecteuclid.org/euclid.aop/1520586273_20180309040445Fri, 09 Mar 2018 04:04 ESTFirst-passage percolation on Cartesian power graphshttps://projecteuclid.org/euclid.aop/1520586274<strong>Anders Martinsson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1004--1041.</p><p><strong>Abstract:</strong><br/>
We consider first-passage percolation on the class of “high-dimensional” graphs that can be written as an iterated Cartesian product $G\square G\square\dots\square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v,v,\dots,v)$ and $(w,w,\dots,w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^{*}_{G}(v,w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^{n}$ as $n\rightarrow\infty$ for a large class of distributions of passage times.
</p>projecteuclid.org/euclid.aop/1520586274_20180309040445Fri, 09 Mar 2018 04:04 ESTSPDE limit of the global fluctuations in rank-based modelshttps://projecteuclid.org/euclid.aop/1520586275<strong>Praveen Kolli</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1042--1069.</p><p><strong>Abstract:</strong><br/>
We consider systems of diffusion processes (“particles”) interacting through their ranks (also referred to as “rank-based models” in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof, we also derive quantitative propagation of chaos estimates for the particle system.
</p>projecteuclid.org/euclid.aop/1520586275_20180309040445Fri, 09 Mar 2018 04:04 ESTExponentially concave functions and a new information geometryhttps://projecteuclid.org/euclid.aop/1520586276<strong>Soumik Pal</strong>, <strong>Ting-Kam Leonard Wong</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1070--1113.</p><p><strong>Abstract:</strong><br/>
A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper, we showed that gradient maps of exponentially concave functions provide solutions to a Monge–Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have an interesting application of determining the optimal trading frequency in stochastic portfolio theory.
</p>projecteuclid.org/euclid.aop/1520586276_20180309040445Fri, 09 Mar 2018 04:04 ESTOn the cycle structure of Mallows permutationshttps://projecteuclid.org/euclid.aop/1520586277<strong>Alexey Gladkich</strong>, <strong>Ron Peled</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1114--1169.</p><p><strong>Abstract:</strong><br/>
We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi\in\mathbb{S}_{n}$ is proportional to $q^{\operatorname{inv}(\pi)}$ where $q>0$ and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$.
We focus on the case that $q<1$ and show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2},n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2}\ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-2}\gg n$ then macroscopic cycles, of size proportional to $n$, emerge. In the second regime, we prove that the distribution of normalized cycle lengths follows the Poisson–Dirichlet law, as in a uniformly random permutation. The results bear formal similarity with a conjectured localization transition for random band matrices.
Further results are presented for the variance of the cycle lengths, the expected diameter of cycles and the expected number of cycles. The proofs rely on the exact sampling algorithm for the Mallows distribution and make use of a special diagonal exposure process for the graph of the permutation.
</p>projecteuclid.org/euclid.aop/1520586277_20180309040445Fri, 09 Mar 2018 04:04 ESTInterlacements and the wired uniform spanning foresthttps://projecteuclid.org/euclid.aop/1520586278<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1170--1200.</p><p><strong>Abstract:</strong><br/>
We extend the Aldous–Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman’s random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of ‘excessive ends’ in the WUSF is nonrandom in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm [ Electron. J. Probab. 13 (2008) 1702–1725], while the third extends a recent result of the author.
Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.
</p>projecteuclid.org/euclid.aop/1520586278_20180309040445Fri, 09 Mar 2018 04:04 ESTGaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processeshttps://projecteuclid.org/euclid.aop/1523520017<strong>Kurt Johansson</strong>, <strong>Gaultier Lambert</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1201--1278.</p><p><strong>Abstract:</strong><br/>
We study mesoscopic linear statistics for a class of determinantal point processes which interpolate between Poisson and random matrix statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the Gaussian Unitary Ensemble (GUE) eigenvalue process. An example of such a system comes from considering the distribution of noncolliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the correlation kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we obtain a central limit theorem (CLT) of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we observe a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.
</p>projecteuclid.org/euclid.aop/1523520017_20180412040031Thu, 12 Apr 2018 04:00 EDTOn global fluctuations for non-colliding processeshttps://projecteuclid.org/euclid.aop/1523520018<strong>Maurice Duits</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1279--1350.</p><p><strong>Abstract:</strong><br/>
We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.
</p>projecteuclid.org/euclid.aop/1523520018_20180412040031Thu, 12 Apr 2018 04:00 EDTMultivariate approximation in total variation, I: Equilibrium distributions of Markov jump processeshttps://projecteuclid.org/euclid.aop/1523520019<strong>A. D. Barbour</strong>, <strong>M. J. Luczak</strong>, <strong>A. Xia</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1351--1404.</p><p><strong>Abstract:</strong><br/>
For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein–Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein’s method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration-death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in $\mathbb{Z}^{d}$.
</p>projecteuclid.org/euclid.aop/1523520019_20180412040031Thu, 12 Apr 2018 04:00 EDTMultivariate approximation in total variation, II: Discrete normal approximationhttps://projecteuclid.org/euclid.aop/1523520020<strong>A. D. Barbour</strong>, <strong>M. J. Luczak</strong>, <strong>A. Xia</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1405--1440.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.aop/1523520020_20180412040031Thu, 12 Apr 2018 04:00 EDTA Gaussian small deviation inequality for convex functionshttps://projecteuclid.org/euclid.aop/1523520021<strong>Grigoris Paouris</strong>, <strong>Petros Valettas</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1441--1454.</p><p><strong>Abstract:</strong><br/>
Let $Z$ be an $n$-dimensional Gaussian vector and let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a convex function. We prove that
\[\mathbb{P}(f(Z)\leq \mathbb{E}f(Z)-t\sqrt{\operatorname{Var}f(Z)})\leq\exp (-ct^{2}),\] for all $t>1$ where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.
</p>projecteuclid.org/euclid.aop/1523520021_20180412040031Thu, 12 Apr 2018 04:00 EDTA variational approach to dissipative SPDEs with singular drifthttps://projecteuclid.org/euclid.aop/1523520022<strong>Carlo Marinelli</strong>, <strong>Luca Scarpa</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1455--1497.</p><p><strong>Abstract:</strong><br/>
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which neither continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.
</p>projecteuclid.org/euclid.aop/1523520022_20180412040031Thu, 12 Apr 2018 04:00 EDTStrong solutions to stochastic differential equations with rough coefficientshttps://projecteuclid.org/euclid.aop/1523520023<strong>Nicolas Champagnat</strong>, <strong>Pierre-Emmanuel Jabin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1498--1541.</p><p><strong>Abstract:</strong><br/>
We study strong existence and pathwise uniqueness for stochastic differential equations in $\mathbb{R}^{d}$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^{p}$ bounds for the solution of the corresponding Fokker–Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, including the uniformly elliptic case in any dimension, the one-dimensional case and the Langevin (kinetic) case.
</p>projecteuclid.org/euclid.aop/1523520023_20180412040031Thu, 12 Apr 2018 04:00 EDTDimensions of random covering sets in Riemann manifoldshttps://projecteuclid.org/euclid.aop/1523520024<strong>De-Jun Feng</strong>, <strong>Esa Järvenpää</strong>, <strong>Maarit Järvenpää</strong>, <strong>Ville Suomala</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1542--1596.</p><p><strong>Abstract:</strong><br/>
Let ${\mathbf{M}}$, ${\mathbf{N}}$ and ${\mathbf{K}}$ be $d$-dimensional Riemann manifolds. Assume that ${\mathbf{A}}:=(A_{n})_{n\in{\mathbb{N}}}$ is a sequence of Lebesgue measurable subsets of ${\mathbf{M}}$ satisfying a necessary density condition and ${\mathbf{x}}:=(x_{n})_{n\in{\mathbb{N}}}$ is a sequence of independent random variables, which are distributed on ${\mathbf{K}}$ according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\mathbf{E}}({\mathbf{x}},{\mathbf{A}}):=\limsup_{n\to\infty}A_{n}(x_{n})\subset{\mathbf{N}}$. Here, $A_{n}(x_{n})$ is a diffeomorphic image of $A_{n}$ depending on $x_{n}$. We also verify that the packing dimensions of ${\mathbf{E}}({\mathbf{x}},{\mathbf{A}})$ equal $d$ almost surely.
</p>projecteuclid.org/euclid.aop/1523520024_20180412040031Thu, 12 Apr 2018 04:00 EDTOptimal surviving strategy for drifted Brownian motions with absorptionhttps://projecteuclid.org/euclid.aop/1523520025<strong>Wenpin Tang</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1597--1650.</p><p><strong>Abstract:</strong><br/>
We study the “Up the River” problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on $\mathbb{R}_{+}$, which are annihilated once they reach the origin. Starting $K$ particles at $x=1$, we prove Aldous’ conjecture [Aldous (2002)] that the “push-the-laggard” strategy of distributing the drift asymptotically (as $K\to\infty$) maximizes the total number of surviving particles, with approximately $\frac{4}{\sqrt{\pi}}\sqrt{K}$ surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-phase partial differential equation (PDE) with a moving boundary, by utilizing certain integral identities and coupling techniques.
</p>projecteuclid.org/euclid.aop/1523520025_20180412040031Thu, 12 Apr 2018 04:00 EDTDiscretisations of rough stochastic PDEshttps://projecteuclid.org/euclid.aop/1523520026<strong>M. Hairer</strong>, <strong>K. Matetski</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1651--1709.</p><p><strong>Abstract:</strong><br/>
We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical $\Phi^{4}_{3}$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the $\Phi^{4}_{3}$ measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.
</p>projecteuclid.org/euclid.aop/1523520026_20180412040031Thu, 12 Apr 2018 04:00 EDTMultidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potentialhttps://projecteuclid.org/euclid.aop/1523520027<strong>Giuseppe Cannizzaro</strong>, <strong>Khalil Chouk</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1710--1763.</p><p><strong>Abstract:</strong><br/>
We study the existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock–Varadhan martingale problem associated to such equations. The approach we exploit is the one of paracontrolled distributions introduced in ( Forum Math. Pi 3 (2015) e6). As a result, we make sense of the three-dimensional polymer measure with white noise potential.
</p>projecteuclid.org/euclid.aop/1523520027_20180412040031Thu, 12 Apr 2018 04:00 EDTChaining, interpolation and convexity II: The contraction principlehttps://projecteuclid.org/euclid.aop/1523520028<strong>Ramon van Handel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1764--1805.</p><p><strong>Abstract:</strong><br/>
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 multiscale 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.
</p>projecteuclid.org/euclid.aop/1523520028_20180412040031Thu, 12 Apr 2018 04:00 EDTLimit theorems for Markov walks conditioned to stay positive under a spectral gap assumptionhttps://projecteuclid.org/euclid.aop/1528876816<strong>Ion Grama</strong>, <strong>Ronan Lauvergnat</strong>, <strong>Émile Le Page</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1807--1877.</p><p><strong>Abstract:</strong><br/>
Consider a Markov chain $(X_{n})_{n\geq0}$ with values in the state space $\mathbb{X}$. Let $f$ be a real function on $\mathbb{X}$ and set $S_{n}=\sum_{i=1}^{n}f(X_{i})$, $n\geq1$. Let $\mathbb{P}_{x}$ be the probability measure generated by the Markov chain starting at $X_{0}=x$. For a starting point $y\in\mathbb{R}$, denote by $\tau_{y}$ the first moment when the Markov walk $(y+S_{n})_{n\geq1}$ becomes nonpositive. Under the condition that $S_{n}$ has zero drift, we find the asymptotics of the probability $\mathbb{P}_{x}(\tau_{y}>n)$ and of the conditional law $\mathbb{P}_{x}(y+S_{n}\leq \cdot\sqrt{n}\mid\tau_{y}>n)$ as $n\to+\infty$.
</p>projecteuclid.org/euclid.aop/1528876816_20180613040038Wed, 13 Jun 2018 04:00 EDTThe fourth moment theorem on the Poisson spacehttps://projecteuclid.org/euclid.aop/1528876817<strong>Christian Döbler</strong>, <strong>Giovanni Peccati</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1878--1916.</p><p><strong>Abstract:</strong><br/>
We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result—that has been elusive for several years—shows that the so-called ‘fourth moment phenomenon’, first discovered by Nualart and Peccati [ Ann. Probab. 33 (2005) 177–193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [ Ann. Probab. 40 (2012) 2439–2459] and Azmoodeh, Campese and Poly [ J. Funct. Anal. 266 (2014) 2341–2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.
</p>projecteuclid.org/euclid.aop/1528876817_20180613040038Wed, 13 Jun 2018 04:00 EDTAnchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaceshttps://projecteuclid.org/euclid.aop/1528876818<strong>Itai Benjamini</strong>, <strong>Elliot Paquette</strong>, <strong>Joshua Pfeffer</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1917--1956.</p><p><strong>Abstract:</strong><br/>
We show that a random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson–Voronoi tessellation and the hyperbolic Poisson–Delaunay triangulation, have $1$-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive hyperbolic speed. Finally, we include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson–Voronoi tessellation.
</p>projecteuclid.org/euclid.aop/1528876818_20180613040038Wed, 13 Jun 2018 04:00 EDTOptimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noisehttps://projecteuclid.org/euclid.aop/1528876819<strong>Viorel Barbu</strong>, <strong>Michael Röckner</strong>, <strong>Deng Zhang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1957--1999.</p><p><strong>Abstract:</strong><br/>
We analyze the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schrödinger equation perturbed by a linear multiplicative Wiener process. The existence of an open-loop optimal control and first-order Lagrange optimality conditions are derived, via Skorohod’s representation theorem, Ekeland’s variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.
</p>projecteuclid.org/euclid.aop/1528876819_20180613040038Wed, 13 Jun 2018 04:00 EDTRandom partitions of the plane via Poissonian coloring and a self-similar process of coalescing planar partitionshttps://projecteuclid.org/euclid.aop/1528876820<strong>David Aldous</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2000--2037.</p><p><strong>Abstract:</strong><br/>
Plant differently colored points in the plane; then let random points (“Poisson rain”) fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In the course of the proof, which heavily exploits time-reversals, we encounter a novel self-similar process of coalescing planar partitions. In this process, sets $A(z)$ in the partition are associated with Poisson random points $z$, and the dynamics are as follows. Points are deleted randomly at rate $1$; when $z$ is deleted, its set $A(z)$ is adjoined to the set $A(z^{\prime})$ of the nearest other point $z^{\prime}$.
</p>projecteuclid.org/euclid.aop/1528876820_20180613040038Wed, 13 Jun 2018 04:00 EDTScaling limit of two-component interacting Brownian motionshttps://projecteuclid.org/euclid.aop/1528876821<strong>Insuk Seo</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2038--2063.</p><p><strong>Abstract:</strong><br/>
This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain form of singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and the interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a partial differential equation known as the Maxwell–Stefan equation.
</p>projecteuclid.org/euclid.aop/1528876821_20180613040038Wed, 13 Jun 2018 04:00 EDTLarge excursions and conditioned laws for recursive sequences generated by random matriceshttps://projecteuclid.org/euclid.aop/1528876822<strong>Jeffrey F. Collamore</strong>, <strong>Sebastian Mentemeier</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2064--2120.</p><p><strong>Abstract:</strong><br/>
We study the large exceedance probabilities and large exceedance paths of the recursive sequence $V_{n}=M_{n}V_{n-1}+Q_{n}$, where $\{(M_{n},Q_{n})\}$ is an i.i.d. sequence, and $M_{1}$ is a $d\times d$ random matrix and $Q_{1}$ is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for $\{V_{n}\}$ and a Cramér-type condition for $\{M_{n}\}$. Under these assumptions, we characterize the distribution of the first passage time $T_{u}^{A}:=\inf\{n:V_{n}\in uA\}$, where $A$ is a general subset of $\mathbb{R}^{d}$, exhibiting that $T_{u}^{A}/u^{\alpha}$ converges to an exponential law for a certain $\alpha>0$. In the process, we revisit and refine classical estimates for $\mathbb{P}(V\in uA)$, where $V$ possesses the stationary law of $\{V_{n}\}$. Namely, for $A\subset\mathbb{R}^{d}$, we show that $\mathbb{P}(V\in uA)\sim C_{A}u^{-\alpha}$ as $u\to\infty$, providing, most importantly, a new characterization of the constant $C_{A}$. As a simple consequence of these estimates, we also obtain an expression for the extremal index of $\{|V_{n}|\}$. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that $\{V_{n}\}$ follows an exponentially-shifted Markov random walk, which we identify. We thereby generalize results from the theory of classical random walk to multivariate recursive sequences.
</p>projecteuclid.org/euclid.aop/1528876822_20180613040038Wed, 13 Jun 2018 04:00 EDTPhase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like treeshttps://projecteuclid.org/euclid.aop/1528876823<strong>Daniel Kious</strong>, <strong>Vladas Sidoravicius</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2121--2133.</p><p><strong>Abstract:</strong><br/>
In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_{0}$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_{0}$ and almost surely transient if $a<a_{0}$. This provides the first examples of phase transition for the Once-reinforced random walk.
</p>projecteuclid.org/euclid.aop/1528876823_20180613040038Wed, 13 Jun 2018 04:00 EDTThe Brownian limit of separable permutationshttps://projecteuclid.org/euclid.aop/1528876824<strong>Frédérique Bassino</strong>, <strong>Mathilde Bouvel</strong>, <strong>Valentin Féray</strong>, <strong>Lucas Gerin</strong>, <strong>Adeline Pierrot</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2134--2189.</p><p><strong>Abstract:</strong><br/>
We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a “Brownian separable permuton”.
</p>projecteuclid.org/euclid.aop/1528876824_20180613040038Wed, 13 Jun 2018 04:00 EDTCritical density of activated random walks on transitive graphshttps://projecteuclid.org/euclid.aop/1528876825<strong>Alexandre Stauffer</strong>, <strong>Lorenzo Taggi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2190--2220.</p><p><strong>Abstract:</strong><br/>
We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $\mu_{c}$ for sustained activity is strictly between 0 and 1. It was known that $\mu_{c}>0$ on $\mathbb{Z}^{d}$, $d\geq1$, and that $\mu_{c}<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_{c}\to0$ as $\lambda\to0$ in all vertex-transitive transient graphs, implying that $\mu_{c}<1$ for small enough sleeping rate. We also show that $\mu_{c}<1$ for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_{c}>0$ in any vertex-transitive amenable graph, and that $\mu_{c}\in(0,1)$ for any sleeping rate on regular trees.
</p>projecteuclid.org/euclid.aop/1528876825_20180613040038Wed, 13 Jun 2018 04:00 EDTIndistinguishability of the components of random spanning forestshttps://projecteuclid.org/euclid.aop/1528876826<strong>Ádám Timár</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2221--2242.</p><p><strong>Abstract:</strong><br/>
We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying “weak insertion tolerance”, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.
</p>projecteuclid.org/euclid.aop/1528876826_20180613040038Wed, 13 Jun 2018 04:00 EDTWeak symmetric integrals with respect to the fractional Brownian motionhttps://projecteuclid.org/euclid.aop/1528876827<strong>Giulia Binotto</strong>, <strong>Ivan Nourdin</strong>, <strong>David Nualart</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2243--2267.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the $\nu$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(\nu)\geq1$ is the largest natural number satisfying $\int_{0}^{1}\alpha^{2j}\nu(d\alpha)=\frac{1}{2j+1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.
</p>projecteuclid.org/euclid.aop/1528876827_20180613040038Wed, 13 Jun 2018 04:00 EDTOn the spectral radius of a random matrix: An upper bound without fourth momenthttps://projecteuclid.org/euclid.aop/1528876828<strong>Charles Bordenave</strong>, <strong>Pietro Caputo</strong>, <strong>Djalil Chafaï</strong>, <strong>Konstantin Tikhomirov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2268--2286.</p><p><strong>Abstract:</strong><br/>
Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance. In other words, that there are no outliers in the circular law. In this work, we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.
</p>projecteuclid.org/euclid.aop/1528876828_20180613040038Wed, 13 Jun 2018 04:00 EDTStochastic Airy semigroup through tridiagonal matriceshttps://projecteuclid.org/euclid.aop/1528876829<strong>Vadim Gorin</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2287--2344.</p><p><strong>Abstract:</strong><br/>
We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_{\beta}$ process, which describes the largest eigenvalues in the $\beta$ ensembles of random matrix theory. Another consequence is a Feynman–Kac formula for the stochastic Airy operator of Edelman–Sutton and Ramirez–Rider–Virag.
As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.
</p>projecteuclid.org/euclid.aop/1528876829_20180613040038Wed, 13 Jun 2018 04:00 EDTOn the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraintshttps://projecteuclid.org/euclid.aop/1528876830<strong>Natesh S. Pillai</strong>, <strong>Aaron Smith</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2345--2399.</p><p><strong>Abstract:</strong><br/>
Determining the total variation mixing time of Kac’s random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with independent entries to our setting. These bounds imply that the mixing time of Kac’s walk on the group $\mathrm{SO}(n)$ is between $C_{1}n^{2}$ and $C_{2}n^{4}\log(n)$ for some explicit constants $0<C_{1},C_{2}<\infty$, substantially improving on the bound of $O(n^{5}\log(n)^{2})$ in the preprint of Jiang [Jiang (2012)]. Our methods may also be applied to other high dimensional Gibbs samplers with constraints, and thus are of independent interest. In addition to giving analytical bounds on the mixing time, our approach allows us to compute rigorous estimates of the mixing time by simulating the eigenvalues of a random matrix.
</p>projecteuclid.org/euclid.aop/1528876830_20180613040038Wed, 13 Jun 2018 04:00 EDTErrata to “Distance covariance in metric spaces”https://projecteuclid.org/euclid.aop/1528876831<strong>Russell Lyons</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2400--2405.</p><p><strong>Abstract:</strong><br/>
We correct several statements and proofs in our paper, Ann. Probab. 41 , no. 5 (2013), 3284–3305.
</p>projecteuclid.org/euclid.aop/1528876831_20180613040038Wed, 13 Jun 2018 04:00 EDT