Malliavin-Stein method: a survey of some recent developments

DOI10.15559/21-VMSTA184zbMath1476.60044arXiv1809.01912MaRDI QIDQ2239800

Ehsan Azmoodeh, Xiaochuan Yang, Giovanni Peccati

Publication date: 5 November 2021

Published in: Modern Stochastics. Theory and Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1809.01912

zbMATH Keywords

weak convergenceMalliavin calculusspectrumStein's methodmultiple integraleigenfunctionlimit theoremsprobability metricseigenspacePoisson spaceWiener spaceMarkov generatorfourth moment theoremsMarkov tripleBerry-Essen boundsfunctional \(\Gamma \)-calculus

Mathematics Subject Classification ID

Central limit and other weak theorems (60F05) Markov semigroups and applications to diffusion processes (47D07) Spectrum, resolvent (47A10) Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators (34L10) Stochastic calculus of variations and the Malliavin calculus (60H07) Convergence of probability measures (60B10) Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) (28C20)

Related Items (2)

On algebraic Stein operators for Gaussian polynomials ⋮ Optimal variance-Gamma approximation on the second Wiener chaos

Cites Work

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• Stochastic analysis for Poisson point processes. Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry
• Generalization of the Nualart-Peccati criterion
• Convergence in law in the second Wiener/Wigner chaos
• Random geometric complexes in the thermodynamic regime
• On Stein's method for infinite-dimensional Gaussian approximation in abstract Wiener spaces
• Chaos of a Markov operator and the fourth moment condition
• Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to \(U\)-statistics and stochastic geometry
• Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization
• Stein's method for rough paths
• Stein characterizations for linear combinations of gamma random variables
• Noncentral convergence of multiple integrals
• Stein's method on Wiener chaos
• Malliavin-Stein method for variance-gamma approximation on Wiener space
• Fluctuations of eigenvalues and second order Poincaré inequalities
• Stein's method and normal approximation of Poisson functionals
• Wiener chaos: Moments, cumulants and diagrams. A survey with computer implementation
• Cumulants on the Wiener space
• Second order Poincaré inequalities and CLTs on Wiener space
• A functional combinatorial central limit theorem
• Stein's method for diffusion approximations
• The fourth moment theorem on the Poisson space
• A Stein deficit for the logarithmic Sobolev inequality
• Quantitative limit theorems for local functionals of arithmetic random waves
• Normal approximation for stabilizing functionals
• Fourth moment theorems on the Poisson space in any dimension
• The Gamma Stein equation and noncentral de Jong theorems
• Gaussian limits for random measures in geometric probability
• An algebra of Stein operators
• The central limit theorem for weighted minimal spanning trees on random points
• Central limit theorems for some graphs in computational geometry.
• Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos
• A bound on the Wasserstein-2 distance between linear combinations of independent random variables
• Approximation of Hilbert-valued gaussians on Dirichlet structures
• Stein's method of exchangeable pairs in multivariate functional approximations
• Donsker's theorem in Wasserstein-1 distance
• Continuous Breuer-Major theorem: tightness and nonstationarity
• Stein's method for multivariate Brownian approximations of sums under dependence
• Stein's method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem
• Non-universal fluctuations of the empirical measure for isotropic stationary fields on \(\mathbb{S}^2\times \mathbb{R} \)
• Entropy and the fourth moment phenomenon
• Malliavin and Dirichlet structures for independent random variables
• Multivariate second order Poincaré inequalities for Poisson functionals
• On central limit theorems in stochastic geometry for add-one cost stabilizing functionals
• The functional Breuer-Major theorem
• An improved second-order Poincaré inequality for functionals of Gaussian fields
• Wasserstein and Kolmogorov error bounds for variance-gamma approximation via Stein's method. I
• Gaussian random measures generated by Berry's nodal sets
• Berry-Esseen bounds in the Breuer-major CLT and Gebelein's inequality
• Existence of Stein kernels under a spectral gap, and discrepancy bounds
• Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one
• Stein kernels and moment maps
• Stein's method, logarithmic Sobolev and transport inequalities
• Minimal spanning trees and Stein's method
• Nodal statistics of planar random waves
• Four moments theorems on Markov chaos
• Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws
• Variance-gamma approximation via Stein's method
• Fourth moment theorems for Markov diffusion generators
• Central limit theorems for multiple stochastic integrals and Malliavin calculus
• Central limit theorems for sequences of multiple stochastic integrals
• Multivariate spatial central limit theorems with applications to percolation and spatial graphs
• The Stein-Dirichlet-Malliavin method
• Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach
• Stein's method, Malliavin calculus, Dirichlet forms and the fourth moment theorem
• Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants
• Lectures on Gaussian Approximations with Malliavin Calculus
• Normal Approximations with Malliavin Calculus
• Random nodal lengths and Wiener chaos
• The Malliavin Calculus and Related Topics
• Normal Approximation by Stein’s Method
• Weak convergence on Wiener space: targeting the first two chaoses
• A quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of S2
• Analysis and Geometry of Markov Diffusion Operators
• Fluctuations of nodal sets on the 3-torus and general cancellation phenomena
• Optimal Gamma Approximation on Wiener Space
• The optimal fourth moment theorem
• Lectures on the Poisson Process
• Optimal Transport
• Stochastic Analysis for Poisson Processes

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