Showing papers in "Electronic Journal of Probability in 2015"
TL;DR: The pseudo spectral gap as discussed by the authors generalises the multiplicative reversiblication approach of Jim Fill and shows that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains.
Abstract: We prove a version of McDiarmid’s bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the “pseudo spectral gap”, and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.
163 citations
TL;DR: The procedure is modified to obtain upper bounds for all p-th moments of the supremum of a stochastic process and gives a significantly simplified proof of the restricted isometry property of the subsampled discrete Fourier transform.
Abstract: We modify Talagrand's generic chaining method to obtain upper bounds for all p-th moments of the supremum of a stochastic process. These bounds lead to an estimate for the upper tail of the supremum with optimal deviation parameters. We apply our procedure to improve and extend some known deviation inequalities for suprema of unbounded empirical processes and chaos processes. As an application we give a significantly simplified proof of the restricted isometry property of the subsampled discrete Fourier transform.
157 citations
TL;DR: In this article, the authors studied the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich.
Abstract: This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Holder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.
141 citations
TL;DR: In this article, the characteristic polynomial of Haar distributed random unitary matrices is studied, and it is shown that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the Polynomial and the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers.
Abstract: We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.
109 citations
TL;DR: In this article, a new version of Stein's method of exchangeable pairs is proposed, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous distribution.
Abstract: We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous distribution. This distribution is characterized by a first order linear differential Stein operator, whose coefficients $\gamma$ and $\eta$ are motivated by two regression properties satisfied by the pair $(W,W')$. Furthermore, the general theory of Stein's method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given. This abstract approach is a certain extension of the theory developed in previous works, which only consider the framework of the density approach, i.e. $\eta\equiv1$. As an illustration of our technique we prove a general plug-in result, which bounds a certain distance of the distribution of a given random variable $W$ to a Beta distribution in terms of a given exchangeable pair $(W,W')$ and provide new bounds on the solution to the Stein equation for the Beta distribution, which complement the existing bounds from previous works. The abstract plug-in result is then applied to derive bounds of order $n^{-1}$ for the distance between the distribution of the relative number of drawn red balls after $n$ drawings in a Polya urn model and the limiting Beta distribution measured by a certain class of smooth test functions.
74 citations
TL;DR: In this article, the partition function of a random polymer model with gamma-distributed weights was shown to be asymptotic in terms of Whittaker functions, and its law can be expressed as the smallest eigenvalue in the Laguerre unitary ensemble.
Abstract: We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppalainen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.
63 citations
TL;DR: In this paper, the authors combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process.
Abstract: We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.
60 citations
TL;DR: In this article, the authors established general exponential inequalities for supermartingales and applied them to linear regressions, autoregressive processes, branching processes, and self-normalized deviations.
Abstract: The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Pena, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of de la Pena’s inequality to self-normalized deviations is also provided.
60 citations
TL;DR: The general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees are proved.
Abstract: We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein' ...
56 citations
TL;DR: In this article, the authors studied the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap and obtained some bounds on the asymptotic behavior when the number of points tends to infinity.
Abstract: The spherical ensemble is a well-studied determinantal process with a fixed number of points on $\mathbb{S}^2$. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two "metrics" to measure the uniformity of anarrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).
53 citations
TL;DR: In this article, the authors studied a random walk in a one-dimensional dynamic random environment and showed that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided the deviation bound is large enough.
Abstract: In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $\rho \in (0,\infty)$. At each step the random walk performs a nearest-neighbour jump, moving to the right with probability $p_{\circ}$ when it is on a vacant site and probability $p_{\bullet}$ when it is on an occupied site. Assuming that $p_\circ \in (0,1)$ and $p_\bullet
eq \tfrac12$, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided $\rho$ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.
TL;DR: Ichiba, Karatzas, Shkolnikov as mentioned in this paper showed a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions in the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them.
Abstract: Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. A triple collision occurs if three particles are at the same position at the same moment. A simultaneous collision occurs if at a certain moment, there are two distinct pairs of particles such that in each pair, both particles occupy the same position. These two pairs of particles can overlap, so a triple collision is a particular case of a simultaneous collision. We find a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions, continuing the work of Ichiba, Karatzas, Shkolnikov (2013). Our results are also valid for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them; these systems were introduced in Karatzas, Pal, Shkolnikov
TL;DR: In this article, the authors consider a branching particle system consisting of particles moving according to an Ornstein-Uhlenbeck process and undergoing binary, supercritical branching with a constant rate.
Abstract: In this paper we consider a branching particle system consisting of particles moving according to an Ornstein-Uhlenbeck process in $\mathbb{R}d$ and undergoing binary, supercritical branching with a constant rate $\lambda > 0$. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalization fall into three qualitatively different classes. In what we call the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalization is the square root of the size of the system. In the critical case the limit is still Gaussian, but the normalization requires an additional term. Finally, when branching has a large rate the situation is completely different. The limit is no longer Gaussian, the normalization is substantially larger than the classical one and the convergence holds in probability. We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process).
TL;DR: In this article, the authors studied a branching random walk in an environment which depends on the time and proved that the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.
Abstract: In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.
TL;DR: In this paper, the authors consider a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle of a Riemannian manifold, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $\sigma$ quantifying the size of the noise.
Abstract: We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$ , collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $\sigma$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ paths in $\mathcal M$ . We show, both qualitively and quantitatively, that the laws of these $\mathcal M$ -valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter $\sigma$ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when $\sigma$ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.
TL;DR: In this article, the authors provided some on-off type criteria for recurrence of regime switching diffusion processes using the theory of M-matrix, the Perron-Frobenius theorem.
Abstract: We provide some on-off type criteria for recurrence of regime-switching diffusion processes using the theory of M-matrix, the Perron-Frobenius theorem. State-independent and state-dependent regime-switching diffusion processes in a finite space and an infinite countable space are both studied. Especially, we put forward a finite partition method to deal with switching process in an infinite countable space. As an application, we study the recurrence of regime-switching Ornstein-Uhlenbeck process, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.
TL;DR: In this article, a strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter γ was proved.
Abstract: We prove a strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter γ. First, we establish that if the asymptotic velocity of the walker is non-zero in the limiting case “γ = ∞", where the environment gets fully refreshed between each step of the walker, then, for γ large enough, the walker still has a non-zero asymptotic velocity in the same direction. Second, we establish that if the walker is transient in the limiting case γ = 0, then, for γ small enough but positive, the walker has a non-zero asymptotic velocity in the direction of the transience. These two limiting velocities can sometimes be of opposite sign. In all cases, we show that the fluctuations are normal.
TL;DR: In this paper, the hitting times of Markov processes to target set G, starting from a reference configuration x0 or its basin of attraction, were studied and the relation to metastability was discussed.
Abstract: We study the hitting times of Markov processes to target set G, starting from a reference configuration x0 or its basin of attraction and we discuss its relation to metastability. Three types of results are reported: (1) A general theory is developed, based on the path-wise approach to metastability, which is general in that it does not assume reversibility of the process, does not focus only on hitting times to rare events and does not assume a particular starting measure. We consider only the natural hypothesis that the mean hitting time to G is asymptotically longer than the mean recurrence time to the refernce configuration x0 or G. Despite its mathematical simplicity, the approach yields precise and explicit bounds on the corrections to exponentiality. (2) We compare and relate different metastability conditions proposed in the literature. This is specially relevant for evolutions of infinite-volume systems. (3) We introduce the notion of early asymptotic exponential behavior to control time scales asymptotically smaller than the mean-time scale. This control is particularly relevant for systems with unbounded state space where nucleations leading to exit from metastability can happen anywhere in the volume. We provide natural sufficient conditions on recurrence times for this early exponentiality to hold and show that it leads to estimations of probability density functions.
TL;DR: In this article, it was shown that it is possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times, and the local time is constructed from discrete approximations.
Abstract: Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is $\alpha$-H\"older continuous for all $\alpha<1/2$. Additionally, we provide various generalizations of Follmer's pathwise Ito formula.
TL;DR: In this paper, a general sufficient condition for a sequence of Galton-Watson branching processes in varying environments to converge weakly is established, which extends previ- ous results by allowing offspring distributions to have infinite variance.
Abstract: We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly This condition extends previ- ous results by allowing offspring distributions to have infinite variance Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes
TL;DR: Local Lyapunov functions are identified for several classes of nonlinear ODEs, including those associated with systems with slow adaptation and Gibbs systems as mentioned in this paper, where positive definite subsolutions of this partial differential equation (PDE) are shown to serve as local LYP functions for the ODE.
Abstract: The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state weakly interacting N-particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using results from [5] and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.
TL;DR: In this paper, it was shown that the model admits a one-step replica symmetry breaking in spin glass terminology, which implies Poisson-Dirichlet statistics by general spin glass arguments.
Abstract: In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions.The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology.This implies Poisson-Dirichlet statistics by general spin glass arguments.In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.
TL;DR: In this paper, a large class of Bernoulli percolation models on random lattices of the half-plane, obtained as local limits of uniform planar triangulations or quadrangulations, were studied.
Abstract: We study a large class of Bernoulli percolation models on random lattices of the half- plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold in the quadrangular case using the so-called peeling techniques. Then, we generalize a result of Angel about the scaling limit of crossing probabilities, that are a natural analogue to Cardy’s formula in (non-random) plane lattices. Our main result is that those probabilities are universal, in the sense that they do not depend on the percolation model neither on the degree of the faces of the map.
TL;DR: In this article, two discretizations of a novel class of Markovian backward stochastic differential equations (BSDEs) are studied, one based on the classical Euler scheme and the other based on Malliavin weights.
Abstract: Two discretizations of a novel class of Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes $Z$, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process $Z$ directly.Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs.These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in the time-grid.The class of BSDEs considered includes Lipschitz BSDEs with fractionally smooth terminal condition as well as quadratic BSDEs with bounded, H\"older continuous terminal condition.
TL;DR: In this article, Bismut's approach about the Malliavin calculus with jumps was used to study the regularity of the distributional density for SDEs driven by degenerate additive Levy noises.
Abstract: By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Levy noises. Under full Hormander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density.Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.
TL;DR: In this article, the authors prove existence of 2-order BSDEs with jumps by a direct method, thus providing complete well-posedness for 2BSDEJ equations, which are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations.
Abstract: In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in an accompanying paper. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.
TL;DR: In this article, a simple construction of the Brownian continuum random tree based on an inhomogeneous Poisson process was given for α = 2, where the lengths of these line-segments are related to the increments of an increasing ε-valued Markov chain.
Abstract: We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $\alpha = 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.
TL;DR: In this article, the authors connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in dimension one.
Abstract: In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.
TL;DR: In this article, the authors considered an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor, and proved convergence to the Airy process for stationary initial configurations.
Abstract: We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit $t\to\infty$. This leads to a new universal cross-over process.
TL;DR: In this article, it was shown that the complex-valued ODE can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term, which has a unique invariant probability measure, which is heavy-tailed yet is uniformly, exponentially attracting.
Abstract: We show that the complex-valued ODE ($n\geq 1$, $a_{n+1}
eq 0$): $$\dot z = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0 , $$ which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant probability measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II extends the main results to the general setting.