Stein's method and exact Berry-Esseen asymptotics for functionals of Gaussian fields

DOI10.1214/09-AOP461zbMath1196.60034arXiv0803.0458MaRDI QIDQ971939

Ivan Nourdin, Giovanni Peccati

Publication date: 17 May 2010

Published in: The Annals of Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0803.0458

zbMATH Keywords

fractional Brownian motionMalliavin calculusquadratic functionalsStein's methodBrownian sheetnormal approximationBerry-Esseen boundsmultiple stochastic integralsoptimal ratesBreuer-Major CLTlocal Edgeworth expansionsToeplitz quadratic forms

Mathematics Subject Classification ID

Gaussian processes (60G15) Central limit and other weak theorems (60F05) Stochastic integrals (60H05) Stochastic calculus of variations and the Malliavin calculus (60H07)

Related Items (50)

Fourth moment bound and stationary Gaussian processes with positive correlation ⋮ Asymptotic behaviours for maximum likelihood estimator of drift parameter in α-Wiener bridge process ⋮ Stein approximation for multidimensional Poisson random measures by third cumulant expansions ⋮ An Edgeworth expansion for functionals of Gaussian fields and its applications ⋮ Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences ⋮ Quantitative clts on a gaussian space: a survey of recent developments ⋮ Stokes formula on the Wiener space and \(n\)-dimensional Nourdin-Peccati analysis ⋮ Berry-Esseen type bound of a sequence \(\left\{\frac{X_N}{Y_N}\right\}\) and its application ⋮ Asymptotics of Yule's nonsense correlation for Ornstein-Uhlenbeck paths: a Wiener chaos approach ⋮ Optimal Berry-Esseen bound for parameter estimation of SPDE with small noise ⋮ Optimal Berry-Esseen bound for an estimator of parameter in the Ornstein-Uhlenbeck process ⋮ Conditional Stein approximation for Itô and Skorohod integrals ⋮ The Berry-Esseen bound for identically distributed random variables by Stein method ⋮ The optimal third moment theorem ⋮ Parameter estimations for the sub-fractional Brownian motion with drift at discrete observation ⋮ High-dimensional central limit theorems for homogeneous sums ⋮ Kolmogorov distance for the central limit theorems of the Wiener chaos expansion and applications ⋮ Convergence rate of CLT for the estimation of Hurst parameter of fractional Brownian motion ⋮ Using Stein's method to analyze Euler-Maruyama approximations of regime-switching jump diffusion processes ⋮ Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra ⋮ Normal Approximation on a Finite Wiener Chaos ⋮ Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos ⋮ On the rate of convergence for central limit theorems of sojourn times of Gaussian fields ⋮ Unnamed Item ⋮ The central limit theorem for cross-variation related to the standard Brownian sheet and Berry-Esseen bounds ⋮ Self-normalized asymptotic properties for the parameter estimation in fractional Ornstein–Uhlenbeck process ⋮ The trace problem for Toeplitz matrices and operators and its impact in probability ⋮ Quantitative fourth moment theorem of functions on the Markov triple and orthogonal polynomials ⋮ Stochastic analysis of Gaussian processes via Fredholm representation ⋮ Optimal Berry-Esseen bound for statistical estimations and its application to SPDE ⋮ Stein's method and normal approximation of Poisson functionals ⋮ Limit theorems for additive functionals of stationary fields, under integrability assumptions on the higher order spectral densities ⋮ Quantitative Breuer-Major theorems ⋮ Cumulants on the Wiener space ⋮ Multivariate normal approximation using Stein's method and Malliavin calculus ⋮ Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos ⋮ Oscillating Gaussian processes ⋮ Berry-Esseen theorems under weak dependence ⋮ Stein's Lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent ⋮ An improved second-order Poincaré inequality for functionals of Gaussian fields ⋮ KOLMOGOROV DISTANCE FOR MULTIVARIATE NORMAL APPROXIMATION ⋮ Third cumulant Stein approximation for Poisson stochastic integrals ⋮ Modeling temporally uncorrelated components of complex-valued stationary processes ⋮ Estimation of local anisotropy based on level sets ⋮ Parameter Estimation for Fractional Ornstein–Uhlenbeck Processes with Discrete Observations ⋮ On rate of convergence in non-central limit theorems ⋮ The optimal fourth moment theorem ⋮ On the area of excursion sets of spherical Gaussian eigenfunctions ⋮ Convergence rate of maximum likelihood estimator of parameter in stochastic partial differential equation ⋮ How does tempering affect the local and global properties of fractional Brownian motion?

Cites Work

• Noncentral convergence of multiple integrals
• Stein's method on Wiener chaos
• Approximations and limit theory for quadratic forms of linear processes
• Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes
• Central limit theorems for non-linear functionals of Gaussian fields
• Berry-Esseen theorems for quadratic forms of Gaussian stationary processes
• Central limit theorems for quadratic forms in random variables having long-range dependence
• On bilinear forms in Gaussian random variables and Toeplitz matrices
• Absolute continuity of probability laws of Wiener functionals
• Semi-martingales et grossissement d'une filtration
• On Toeplitz type quadratic functionals of stationary Gaussian processes
• Stochastic integrals: A combinatorial approach
• Gaussian approximations of multiple integrals
• Central limit theorems for multiple stochastic integrals and Malliavin calculus
• On quadratic functionals of the Brownian sheet and related processes
• Central limit theorems for sequences of multiple stochastic integrals
• A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate
• VALID EDGEWORTH EXPANSION FOR THE SAMPLE AUTOCORRELATION FUNCTION UNDER LONG RANGE DEPENDENCE
• The Malliavin Calculus and Related Topics
• CLT and other limit theorems for functionals of Gaussian processes
• Gaussian Hilbert Spaces
• The bootstrap and Edgeworth expansion
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Stein's method and exact Berry-Esseen asymptotics for functionals of Gaussian fields