Gaussian Random Measures Generated by Berry’s Nodal Sets

We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry’s random wave model to a finite collection of (possibly overlapping) smooth compact subsets of R2. Our main result shows that, as the energy diverges to infinity and after an adequate normalisation, these random elements converge in distribution to a Gaussian vector, whose covariance structure reproduces that of a homogeneous independently scattered random measure. A by-product of our analysis is that, when restricted to rectangles, the dominant chaotic projection of the nodal length field weakly converges to a standard Wiener sheet, in the Banach space of real-valued continuous mappings over a fixed compact set. An analogous study is performed for complex-valued random waves, in which case the nodal set is a locally finite collection of random points

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

Malliavin–stein method: A survey of some recent developments

Initiated around the year 2007, the Malliavin–Stein approach to probabilistic approximations combines Stein’s method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin–Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin–Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.ES (R-AGR-3376-10) at Lux embourg University. Xiaochuan Yang is supported by the EPSRC grant EP/T028653/

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

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

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

Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second version mainly resolves minor LaTeX problems.

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

Stein's method on Wiener chaos

We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-It\^o integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Ess\'een bounds in the Breuer-Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.Comment: 39 pages; Two sections added; To appear in PTR

Molecular Recognition of Glycan-Bearing Glycomacromolecules Presented at Membrane Surfaces by Lectins: An NMR View

Lectin–glycan interactions are at the heart of a multitude of biological events. Glycans are usually presented in a multivalent manner on the cell surface as part of the so-called glycocalyx, where they interact with other entities. This multivalent presentation allows us to overcome the typical low affinities found for individual glycan–lectin interactions. Indeed, the presentation of glycans may drastically impact their binding by lectins, highly affecting the corresponding binding affinity and even selectivity. In this context, we herein present the study of the interaction of a variety of homo- and heteromultivalent lactose-functionalized glycomacromolecules and their lipid conjugates with two human galectins. We have employed as ligands the glycomacromolecules, as well as liposomes decorated with those structures, to evaluate their interactions in a cell-mimicking environment. Key details of the interaction have been unravelled by NMR experiments, both from the ligand and receptor perspectives, complemented by cryo-electron microscopy methods and molecular dynamics simulations.M.H. and L.H. thank the DFG for support through the ViroCarb research consortium (HA5950/5-2) and the CeMSA@HHU (Center for Molecular and Structural Analytics @ Heinrich-Heine University) for recording the mass spectrometric and the NMR-spectroscopic data for the structural conformation of the glycomacromolecules and their lipid conjugates. The CIC bioGUNE EM platform is also thanked for infrastructural support during cryo-EM data collection. The group in Spain thank the European Research Council (RECGLYCANMR, Advanced grant no. 788143), MCIN/AEI/10.13039/501100011033 for grants PDI2021-1237810B-C21, PID2021-126130OB-I00, CEX2021-001136-S, and CIBERES, an initiative of Instituto de Salud Carlos III (ISCIII), Madrid, Spain, for generous funding