Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators $\Gamma_i$ (introduced by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly \cite{n-po-1}, concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times

The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion

The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(\delta^{2H-1})$ ($\delta$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(\delta^H)$.Comment: 21 pages, (incorrect) weak convergence result removed, to appear in Stochastic

Employee benefits and challenges of telecommuting virtual working arrangements in the services industry

M. Comm.Virtual working arrangements, including telecommuting, are on the increase globally due to the challenges that organisations face in the current global economy. Virtual working arrangements present considerable possible benefits to organisations, employees and the community at large if correctly implemented. It is estimated that 45 million Americans teleworked in 2006 alone (O’Brien & Hayden, 2007) with predictions of the number reaching 100 million in the United States of America by 2010 (Wilsker, 2008). However, in South Africa this organisational form is not well documented or implemented presently. As a result, local organisations are unaware of the employee benefits and challenges that will be faced when implementing a telecommuting programme and how best to implement teleworking arrangements with these factors in mind

Two properties of vectors of quadratic forms in Gaussian random variables

We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting consequences. Our second result gives a characterization of limits in law for sequences of such vectors.Comment: 14 page

Lectures on Gaussian approximations with Malliavin calculus

In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the "Fourth Moment Theorem" in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor gave a multidimensional version of this characterization. Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and Ortiz-Latorre, giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper "Stein's method on Wiener chaos" (written in collaboration with Peccati) in which, by bringing together Stein's method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the Fourth Moment Theorem. It turns out that Stein's method and Malliavin calculus fit together admirably well. Their interaction has led to some remarkable new results involving central and non-central limit theorems for functionals of infinite-dimensional Gaussian fields. The current survey aims to introduce the main features of this recent theory. It originates from a series of lectures I delivered at the Coll\`ege de France between January and March 2012, within the framework of the annual prize of the Fondation des Sciences Math\'ematiques de Paris. It may be seen as a teaser for the book "Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality" (jointly written with Peccati), in which the interested reader will find much more than in this short survey.Comment: 72 pages. To be published in the S\'eminaire de Probabilit\'es. Mild update: typos, referee comment

Berry-esseen bounds in the breuer-major CLT and gebelein’s inequality

We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function ϕ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein’s inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

Copyright © The Author(s) 2021. We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.FNR grant APOGee (R-AGR-3585-10) at Luxembourg University; FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University; FNR Grant MISSILe (R-AGR-3410-12-Z) at Luxembourg and Singapore Universities

On Nonlinear Functionals of Random Spherical Eigenfunctions

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.Comment: 24 page

Non-integrable Stable Approximation by Stein’s Method

We develop Stein’s method for α-stable approximation with α ∈ (0, 1], continuing the recent line of research by Xu [40] and Chen, Nourdin and Xu [11] in the case α ∈ (1, 2). The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions, and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution t