Stein’s method and Normal approximation of Poisson functionals

By Giovanni Peccati, J. L. Sol'e, Murad S. Taqqu, Frederic Utzet. 2008

154 citations recorded by the source corpus.

We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Esseen bounds in Central limit theorems (CLTs) involving multiple Wiener-Ito integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck Levy processes.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ba03724a071ae7292e1fa01c6487b9b47cbc4c5e

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Citations in the corpus

Chaos of a Markov operator and the fourth moment condition (2012, background): “Analogs of Stein’s lemma in the context of the preceding statements have been investigated on discrete Poisson or Bernoulli spaces in [16, 21, 24].”
The scaling limit of Poisson-driven order statistics with applications in geometric probability (2012, background): “We will briefly recall their definitions and refer to [21, 23] for further details.”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, methodology): “- Reference [29], by Peccati, Sol´e, Taqqu and Utzet, concerns a combination of Stein’s method with a version of Malliavin calculus on the Poisson space (as developed by Nualart and Vives in [28]).”
Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, methodology): “[11] derived by a combination of Stein’s method and Malliavin calculus the upper bound”
Fourth Moment Theorems for Markov Diffusion Generators (2013, methodology): “• Due to the simplicity of our proofs which are only of spectral nature, we believe that our strategy can also be applied in the free and discrete settings (see [3] for Wigner and [14] for Poisson chaos).”
Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, methodology): “In the proofs we use the Malliavin-Stein method for Poisson functionals that was recently developed in [21, 23].”
Lectures on the Poisson Process (2017, methodology): “The normal approximation of higher order stochastic integrals and Poisson U-statistics was treated in [124] and in Reitzner and Schulte [135].”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “2) our results provide substantial extensions of the findings contained in [16, 36, 39], which only considered linear and quadratic functionals.”
Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, methodology): “In [17] and [18], the authors have proved the following limit theorem for Q(T, λ).”
Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry (2014, background): “19) in [29] and of the relation stated in [33], Lemma 2.”
Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs (2011, methodology): “Our main theoretical tools come from the two papers [26, 27], respectively by Peccati et al. and Peccati and Zheng, where the normal approximation of functional of Poisson measures is studied by combining two probabilistic techniques, namely the Stein’s method and the Malliavin calculus of…”
Poisson process Fock space representation, chaos expansion and covariance inequalities (2009, background): “Additional results in Section 3 are concerned with certain derivative and integral operators which are important in Malliavin calculus on Poisson spaces (see for example [32]).”
Moments and Central Limit Theorems for Some Multivariate Poisson Functionals (2012, methodology): “Univariate central limit theorems with bounds on theWasserstein distance for Poisson functionals with finite Wiener–Itô chaos expansions and, in particular, Poisson U statistics are derived in [8], [9], and [16] using a general result from [14].”
Stein's Method and Stochastic Analysis of Rademacher Functionals (2008, methodology): “Technical proofs are deferred to the Appendix.”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “References [21] and [15] use appropriate versions of the non-Gaussian Malliavin calculus in order to deal with the one-dimensional normal approximation, respectively, of functionals of Poisson measures and of functionals of infinite Rademacher sequences.”
Central limit theorems for $U$-statistics of Poisson point processes (2011, methodology): “[27], we show convergence in the Wasserstein distance.”