Malliavin Operators Research Articles

The Breuer–Major theorem in total variation: Improved rates under minimal regularity

In this paper we prove an estimate for the total variation distance, in the framework of the Breuer–Major theorem, using the Malliavin–Stein method, assuming the underlying function g to be once weakly differentiable with g and g′ having finite moments of order four with respect to the standard Gaussian density. This result is proved by a combination of Gebelein’s inequality and some novel estimates involving Malliavin operators.

Stable limit theorems on the Poisson space

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Wasserstein transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by G. Peccati, J. L. Sole, M. S. Taqqu & F. Utzet. “Stein’s method and normal approximation of Poisson functionals” for Gaussian approximations; and by G. Peccati “The Chen-Stein method for Poisson functionals” for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of C. Dobler & G.Peccati “The fourth moment theorem on the Poisson space”. We give an application to stochastic processes.

Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity λ>0. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of O(λ−1∕4) for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.

Malliavin calculus for generalized and test stochastic processes

We extend the Malliavin calculus from the classical finite variance setting to generalized processes with infinite variance and their test processes. The domain and the range of the basic Malliavin operators is characterized in terms of test processes and generalized processes. Various properties are proved such as the duality of the integral and the derivative in strong and in weak sense, the product rule with respect to ordinary and Wick multiplication and the chain rule in classical and in Wick sense.

Quantitative stable limit theorems on the Wiener space

We use Malliavin operators in order to prove quantitative stable limit theorems on the Wiener space, where the target distribution is given by a possibly multidimensional mixture of Gaussian distributions. Our findings refine and generalize previous works by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39-64] and Harnett and Nualart [Stochastic Process. Appl. 122 (2012) 3460-3505], and provide a substantial contribution to a recent line of research, focussing on limit theorems on the Wiener space, obtained by means of the Malliavin calculus of variations. Applications are given to quadratic functionals and weighted quadratic variations of a fractional Brownian motion.

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincare inequality, as well as on the use of Malliavin operators, of Stein’s method, and of an (integrated) Mehler’s formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals.

Fundamental equations with higher order Malliavin operators

We consider three fundamental equations of the Malliavin calculus: the equation involving the Malliavin derivative, the Skorokhod integral and the Ornstein–Uhlenbeck operator of order , . These equations provide a complete characterization of the domain and range of the aforementioned operators. Applying the chaos expansion method in white noise spaces we solve these equations and obtain an explicit form of the solutions in the space of Kondratiev generalized stochastic processes.

Malliavin-Stein method for variance-gamma approximation on Wiener space

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.

Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering

Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic independence of U-statistics following Gaussian and Poisson asymptotic regimes. Our results generalize and unify several previous findings in the field. We provide an application to joint sub-graph counting in random geometric graphs.

Stein's Method and Stochastic Analysis of Rademacher Functionals

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. As the bounds are given in terms of Malliavin operators, no coupling construction is required. When the functional depends only on the first d coordinates of the Rademacher sequence, a simple sufficient condition for convergence to a normal distribution is derived. For finite quadratic forms, we obtain necessary and sufficient conditions. Although our approach does not require the classical use of exchangeable pairs, when the functional depends only on the first d coordinates of the Rademacher sequence we employ chaos expansion in order to construct an explicit exchangeable pair vector; the elements of the vector relate to the summands in the chaos decomposition and satisfy a linearity condition for the conditional expectation. Among several examples, such as random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson.

A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus

In many stochastic partial differential equations (SPDEs) involving random coefficients, modeling the randomness by spatial white noise may lead to ill-posed problems. Here we consider an elliptic problem with spatial Gaussian coefficients and present a methodology that resolves this issue. It is based on stochastic convolution implemented via generalized Malliavin operators in conjunction with weighted Wiener spaces that ensure the ellipticity condition. We present theoretical and numerical results that demonstrate the fast convergence of the method in the proper norm. Our approach is general and can be extended to other SPDEs and other types of multiplicative noise.