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/

On algebraic Stein operators for Gaussian polynomials

The first essential ingredient to build up Stein's method for a continuous target distribution is to identify a so-called \textit{Stein operator}, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of \textit{algebraic} Stein operators (see Definition \ref{def:algebraic-Stein-Operator}), and provide a novel algebraic method to find \emph{all} the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form $Y=h(X)$, where $X=(X_1,\dots, X_d)$ has i.i.d$.$ standard Gaussian components and h\in \KK[X] is a polynomial with coefficients in the ring \KK. Our approach links the existence of an algebraic Stein operator with \textit{null controllability} of a certain linear discrete system. A \texttt{MATLAB} code checks the null controllability up to a given finite time $T$ (the order of the differential operator), and provides all \textit{null control} sequences (polynomial coefficients of the differential operator) up to a given maximum degree $m$. This is the first paper that connects Stein's method with computational algebra to find Stein operators for highly complex probability distributions, such as $H_{20}(X_1)$, where $H_p$ is the $p$-th Hermite polynomial. A number of examples of Stein operators for $H_p(X_1)$, $p=3,4,5,6,7,8,10,12$, are gathered in the extended Appendix of this arXiv version. We also introduce a widely applicable approach to proving that Stein operators characterise the target distribution, and use it to prove, amongst other examples, that the Stein operators for $H_p(X_1)$, $p=3,\ldots,8$, with minimum possible maximal polynomial degree $m$ characterise their target distribution.Comment: 47 pages. This is version submitted for publication, and fills in a gap in the proof of Proposition 3.2. Version 1 contains an extended Appendix B of Stein operators for univariate Gaussian polynomials, giving extra examples than are given in this versio

Extremes of Gaussian random fields with regularly varying dependence structure

Let be a centered Gaussian random field with variance function sigma (2)(ai...) that attains its maximum at the unique point , and let . For a compact subset of a"e, the current literature explains the asymptotic tail behaviour of under some regularity conditions including that 1 - sigma(t) has a polynomial decrease to 0 as t -> t (0). In this contribution we consider more general case that 1 - sigma(t) is regularly varying at t (0). We extend our analysis to Gaussian random fields defined on some compact set , deriving the exact tail asymptotics of for the class of Gaussian random fields with variance and correlation functions being regularly varying at t (0). A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics

Internet of Things for Sustainable Community Development: Introduction and Overview

The two-third of the city-dwelling world population by 2050 poses numerous global challenges in the infrastructure and natural resource management domains (e.g., water and food scarcity, increasing global temperatures, and energy issues). The IoT with integrated sensing and communication capabilities has the strong potential for the robust, sustainable, and informed resource management in the urban and rural communities. In this chapter, the vital concepts of sustainable community development are discussed. The IoT and sustainability interactions are explained with emphasis on Sustainable Development Goals (SDGs) and communication technologies. Moreover, IoT opportunities and challenges are discussed in the context of sustainable community development

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

An asymptotic approach to proving sufficiency of Stein characterisations

In extending Stein’s method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For X denoting a standard Gaussian random variable and Hp the p-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for Hp(X), p = 3; 4;:::; 8, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of p = 3; 4;:::; 8 independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases p = 1; 2 are already known to be characterising). We leverage our Stein characterisations of H3(X) and H4(X) to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method

Approximate methods to evaluate storey stiffness and interstory drift of RC buildings in seismic area

During preliminary design of a RC building located in a seismic area, having quick but reliable analytical measurement of interstory drifts and storey stiffnesses might be helpful in order to check the fulfillment of damage limit state and stiffness regularity in elevation required by seismic design codes. This paper presents two approximate methods, strongly interrelated each other, and addressed to achieve each of these two purposes for frame buildings. A brief description of some already existing methods addressed to the same aims is included to compare the main differences in terms of general approaches and assumptions. Both new approximate methods are then applied to 9 ‘ideal’ frames and 2 ‘real’ buildings designed according to the Italian seismic code. The results are compared with the ‘exact’ values obtained by the code-based standard calculation, performed via FEM models, showing a satisfactory range of accuracy. Compared with those by the other methods from literature, they indicate the proposed procedures lead to a better approximation of the objective structural parameters, especially for those buildings designed according to the modern ‘capacity design’ philosophy

On algebraic Stein operators for Gaussian polynomials

The first essential ingredient to build up Stein’s method for a continuous target distribution is to identify a so-called Stein operator, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of algebraic Stein operators (see Definition 3.4), and provide a novel algebraic method to find all the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form Y = h(X), where X = (X1,…,Xd) has i.i.d. standard Gaussian components and h ∈ K[X] is a polynomial with coefficients in the ring K. Our approach links the existence of an algebraic Stein operator with null controllability of a certain linear discrete system. A MATLAB code checks the null controllability up to a given finite time T (the order of the differential operator), and provides all null control sequences (polynomial coefficients of the differential operator) up to a given maximum degree m. This is the first paper that connects Stein’s method with computational algebra to find Stein operators for highly complex probability distributions, such as H20(X1), where Hp is the p-th Hermite polynomial. Some examples of Stein operators for Hp (X1), p = 3,4,5,6, are gathered in the Appendix and many other examples are given in the Supplementary Information

An asymptotic approach to proving sufficiency of Stein characterisations

In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For $X$ denoting a standard Gaussian random variable and $H_p$ the $p$-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for $H_p(X)$, $p=3,4,\ldots,8$, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of $p=3,4,\ldots,8$ independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases $p=1,2$ are already known to be characterising). We leverage our Stein characterisations of $H_3(X)$ and $H_4(X)$ to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method