Lévy area for Gaussian processes: A double Wiener-Itô integral approach

Let {X₁(t)}_0≤t≤1 and {X₂(t)}_0≤t≤1 be two independent continuous centered Gaussian processes with covariance functions R₁ and R₂. We show that if the covariance functions are of finite p-variation and q-variation respectively and such that p⁻¹+q⁻¹>1, then the Lévy area can be defined as a double Wiener-Itô integral with respect to an isonormal Gaussian process induced by X₁ and X₂. Moreover, some properties of the characteristic function of that generalised Lévy area are studied.     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

On smoothness conditions and convergence rates in the CLT in Banach spaces

Summary In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(n^–1/2) for sets which have regular boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRGResearch supported by the SFB 343 at Bielefeld     

On convergence determining and separating classes of functions

Herein, we generalize and extend some standard results on the separation and convergence of probability measures. We use homeomorphism-based methods and work on incomplete metric spaces, Skorokhod spaces, Lusin spaces or general topological spaces. Our contributions are twofold: we dramatically simplify the proofs of several basic results in weak convergence theory and, concurrently, extend these results to apply more immediately in a number of settings, including on Lusin spaces.     

On rates of convergence and asymptotic normality in the multiknapsack problem

In Meanti et al. (1990) an almost sure asymptotic characterization has been derived for the optimal solution value as function of the knapsack capacities, when the profit and requirement coefficients of items to be selected from are random variables. In this paper we establish a rate of convergence for this process using results from the theory of empirical processes.     

Rates of convergence in the central limit theorem for linear statistics of martingale differences

In this paper, we give rates of convergence for minimal distances between linear statistics of martingale differences and the limiting Gaussian distribution. In particular the results apply to the partial sums of (possibly long range dependent) linear processes, and to the least squares estimator in some parametric regression models.     

On the false discovery proportion convergence under Gaussian equi-correlation

We study the convergence of the false discovery proportion (FDP) of the Benjamini-Hochberg procedure in the Gaussian equi-correlated model, when the correlation ρ_m converges to zero as the hypothesis number m grows to infinity. In this model, the FDP converges to the false discovery rate (FDR) at rate {min(m,1/ρ_m)}^1/2, which is different from the standard convergence rate m^1/2 holding under independence.     

Compound Poisson approximation for unbounded functions on a group,with application to large deviations

Summary A second order error bound is obtained for approximating h d by h d , where is a convolution of measures andQ a compound Poisson measure on a measurable abelian group, and the functionh is not necessarily bounded. This error bound is more refined than the usual total variation bound in the sense that it contains the functionh. The method used is inspired by Stein's method and hinges on bounding Radon-Nikodym derivatives related to . The approximation theorem is then applied to obtain a large deviation result on groups, which in turn is applied to multivariate Poisson approximation.Research of the second author was supported by Schweizerischer Nationalfonds     

Population dynamical behavior of Lotka-Volterra system under regime switching

In this paper, we investigate a Lotka-Volterra system under regime switching

     

On complete convergence for weighted sums of asymptotically linear negatively dependent random field

In this paper we establish the complete convergence for weighted sums of asymptotically linear negatively quadrant dependent random field, which contains a linear negatively quadrant dependent field and a ρ^∗${\rho }^{\ast }$-mixing random field.     

Operator self similar stochastic processes in Rd

Operator self similar stochastic processes taking values in a finite dimensional Euclidean space are introduced and some of their properties are studied.     

Pointwise properties of convergence in probability

If a sequence of random variables X_n converges to X in probability we know little about the pointwise behavior X_n(ω). In this note we show that if X_n converges to X quickly enough (for example, like n^?α for α > 0) then, for almost all ω, X_n(ω) converges to X(ω) outside a set of density zero.     

Central limit theorems for multiple stochastic integrals and Malliavin calculus

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177–193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.     

On the rate of convergence of simple and jump-adapted weak Euler schemes for Lévy driven SDEs

The paper studies the rate of convergence of a weak Euler approximation for solutions to possibly completely degenerate SDEs driven by Lévy processes, with Hölder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes and its robustness to the approximation of the increments of the driving process. A convergence rate is derived for some approximate jump-adapted Euler scheme as well.     

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.     

One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation

This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. It is shown that for any positive damping and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random attractor is a one-dimensional random horizontal curve regardless of the strength of noise. Hence its dynamics is not chaotic. It is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large, which implies that the solutions tend to oscillate with the same frequency eventually and the so-called frequency locking is successful.