Showing papers in "Annals of Probability in 2009"

Stationary max-stable fields associated to negative definite functions.

Abstract: Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1∞δUi be a Poisson point process on the real line with intensity e−y dy. We show that the law of the random family of functions {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=⋁i=1∞Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

Abstract: We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609–631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817–840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

Abstract: We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. We study the connection between eigenvalue statistics on microscopic energy scales $\eta\ll1$ and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order $\eta \sim\log N/N$. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales $\eta\gg N^{-2/3}$. We show that most eigenvectors are fully delocalized in the sense that their $\ell^p$-norms are comparable with $N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound $N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.

Fractional Cauchy problems on bounded domains

Abstract: Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

Central limit theorem for linear eigenvalue statistics of random matrices with independent entries

Abstract: We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) κ 4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, oo) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5 ). This is done by using a simple "interpolation trick" from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially ℂ 5 test function. Here the variance of statistics contains an additional term proportional to κ 4 . The proofs of all limit theorems follow essentially the same scheme.

Mean-field backward stochastic differential equations : a limit approach

Abstract: Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

Contact processes on random graphs with power law degree distributions have critical value 0

Abstract: If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λ c of the infection rate is positive if the power α > 3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gomez-Gardenes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399-1404]. Here, we show that the critical value λ c is zero for any value of α > 3, and the contact process starting from all vertices infected, with a probability tending to 1 as n → ∞, maintains a positive density of infected sites for time at least exp(n 1-δ ) for any δ > 0. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability ρ(λ). It is expected that ρ(λ) ~ Cλ β as → 0. Here we show that α - 1 ≤ β ≤ 2α - 3, and so β > 2 for α > 3. Thus even though the graph is locally tree-like, β does not take the mean field critical value β = 1.

Anticipated backward stochastic differential equations

Abstract: In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.

Minima in branching random walks

Abstract: Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\\mathbfEM_n$ to within O(1) and prove exponential tail bounds for $\\mathbfP{|M_n-\\mathbfEM_n|>x}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89―108], our results fully characterize the possible behavior of $\\mathbf EM_n$ when the branching random walk has bounded branching and step size.

Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition

Abstract: In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein’s method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein’s exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.

The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations.

Abstract: We investigate the asymptotic spectrum of complex or real Deformed Wigner matrices when the entries of the Hermitian (resp., symmetric) Wigner matrix have a symmetric law satisfying a Poincare inequality. The perturbation is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of the perturbation are sufficiently far from zero, the corresponding eigenvalues of the deformed Wigner matrix almost surely exit the limiting semicircle compact sup- port as the size becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of the Wigner matrix. On the other hand, when the perturbation is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of the Wigner matrix.

A vector-valued almost sure invariance principle for hyperbolic dynamical systems

Abstract: We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A dieomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.

Existence of strong solutions for stochastic porous media equation under general monotonicity conditions

Abstract: This paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dX ( X)dt = B(X)dW (t) in bounded domains of R d with Dirichlet boundary conditions. Here is a maximal monotone graph in R R (possibly multivalued) with

Fractional diffusion equations and processes with randomly varying time.

Abstract: In this paper the solutions u ν = u ν (x, t) to fractional diffusion equations of order 0 < v ≤ 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order v = 1 2 n , n ≥ 1, we show that the solutions u 1/2 n correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order v = 2 3 n , n ≥ 1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that u ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

Quasi-stationary distributions and diffusion models in population dynamics

Abstract: In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to $- \infty$ at the origin, and the diffusion to have an entrance boundary at $+\infty$. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth--death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability $1$. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near $0$ and near $+ \infty$ for the existence of quasi-stationary distributions, as well as rate of convergence in the Yaglom limit and existence of the $Q$-process. We also show that under these conditions, there is exactly one quasi-stationary distribution, and that this distribution attracts all initial distributions under the conditional evolution, if and only if $+\infty$ is an entrance boundary. In particular this gives a sufficient condition for the uniqueness of quasi-stationary distributions. In the proofs spectral theory plays an important role on $L^2$ of the reference measure for the killed process.

Weighted poincaré-type inequalities for cauchy and other convex measures

Abstract: Brascamp-Lieb-type, weighted Poincare-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

Bootstrap percolation in three dimensions

Abstract: By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices “infected” at time 0, new vertices are subsequently infected, at each time step, if they have at least r∈ℕ previously infected neighbors. When the set A is chosen at random, the main aim is to determine the critical probability pc(G, r) at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the d-dimensional grid [n]d: with 2≤r≤d fixed, it was proved by Cerf and Cirillo (for d=r=3), and by Cerf and Manzo (in general), that $$p_{c}([n]^{d},r)=\Theta\biggl(\frac{1}{\log_{(r-1)}n}\biggr)^{d-r+1}$$ where log(r) is an r-times iterated logarithm. However, the exact threshold function is only known in the case d=r=2, where it was shown by Holroyd to be $(1+o(1))\frac{\pi^{2}}{18\log n}$. In this paper we shall determine the exact threshold in the crucial case d=r=3, and lay the groundwork for solving the problem for all fixed d and r.

From random matrices to random analytic functions

Abstract: We consider two families of random matrix-valued analytic functions: (1) G 1 - zG 2 and (2) G 0 + zG 1 + z 2 G 2 + ···, where G i are n x n random matrices with independent standard complex Gaussian entries. The random set of z where these matrix-analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain Hilbert spaces ("Bargmann-Fock spaces") of holomorphic functions on the corresponding surfaces. Along with the new results, this also gives a unified framework in which to view a theorem of Peres and Virag (n = 1 in the second setting above) and a well-known result of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).

Noncentral convergence of multiple integrals

Abstract: Fix ν>0, denote by G(v/2) a Gamma random variable with parameter v/2, and let n≥2 be a fixed even integer. Consider a sequence (F_k) of square integrable random variables, belonging to the nth Wiener chaos of a given Gaussian process and with variance converging to 2v. As k goes to infinity, we prove that F_k converges in distribution to 2G(v/2)-v if and only if E(F_k^4)-12 E(F_k^3) tends to 12v^2-48v.

On the structure of quasi-stationary competing particle systems

Abstract: We study point processes on the real line whose configurations X are locally finite, have a maximum and evolve through increments which are functions of correlated Gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {q ij ) i,j∈ ℕ. A probability measure on the pair (X, Q) is said to be quasi-stationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasi-stationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchically nested Poisson-Dirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi-stationary. The main result of this work is a proof of this conjecture for the case where q ij assume only a finite number of values. The result is of relevance for mean-field spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchical organization of the Gibbs measure was first proposed as an ansatz.

Variations and estimators for self-similarity parameters via Malliavin calculus.

Abstract: Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$ Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time $1$ can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in(1/2,2/3)$

Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields

Abstract: We show how to detect optimal Berry–Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein’s method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75–118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395–406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551–579]); (ii) to “exploding” quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer–Major CLT for functionals of a fractional Brownian motion.

Entropic measure and Wasserstein diffusion

Abstract: We construct a new random probability measure on the circle and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval, respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin’s Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.

Tightness for a family of recursion equations

Abstract: In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.

Spinal partitions and invariance under re-rooting of continuum random trees

Abstract: We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson-Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

Uniform limit theorems for wavelet density estimators

Abstract: Let pn(y)=∑kαkϕ(y−k)+∑l=0jn−1∑kβlk2l/2ψ(2ly−k) be the linear wavelet density estimator, where ϕ, ψ are a father and a mother wavelet (with compact support), αk, βlk are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p0 on ℝ, and jn∈ℤ, jn↗∞. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup y∈ℝ|pn(y)−Epn(y)| is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup y∈ℝ|pn(y)−p0(y)| attains the optimal almost sure rate of convergence for estimating p0, if jn is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of pn, that is, for the stochastic processes $\sqrt{n}(F_{n}^{W}(s)-F(s))=\sqrt{n}\int_{-\infty}^{s}(p_{n}-p_{0})$, s∈ℝ, are proved; and more generally, uniform central limit theorems for the processes $\sqrt{n}\int(p_{n}-p_{0})f$, $f\in\mathcal{F}$, for other Donsker classes $\mathcal{F}$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508–539].

A two cities theorem for the parabolic Anderson model.

Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation partial derivative(t)u(t, z) = Delta u(t,z) + xi(z)u(t,z) on (0,infinity) x Z(d) with random potential (xi(z): z is an element of Z(d)). We consider independent and identically distributed potentials, such that the distribution function of (z) converges polynomially at infinity. If u is initially localized in the origin, that is, if u(0, z) = 1(0)(z), we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

A characterization of dimension free concentration in terms of transportation inequalities

Abstract: The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincare inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.

Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4

Abstract: We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion B with Hurst index H=1/4. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to B.

Invariant transports of stationary random measures and mass-stationarity

Abstract: We introduce and discuss balancing invariant transports of stationary random measures on a Polish Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu’s exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transports. We then introduce (in a non-stationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary.