The Chen-Stein method for Poisson functionals

2011/12/21 by Giovanni Peccati, Peccati, Giovanni · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1112.5051

openalex publication_date 2011/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We establish a general inequality on the Poisson space, yielding an upper bound for the distance in total variation between the law of a regular random variable with values in the integers and a Poisson distribution. Several applications are provided, in particular: (i) to deduce a set of sufficient conditions implying that a sequence of (suitably shifted) multiple Wiener-Itô integrals converges in distribution to a Poisson random variable, and (ii) to compute explicit rates of convergence for the Poisson approximation of statistics associated with geometric random graphs with sparse connections (thus refining some findings by Lachièze-Rey and Peccati (2011)). This is the first paper studying Poisson approximations on configuration spaces by combining the Malliavin calculus of variations and the Chen-Stein method.

Citations

• Exact and asymptotic results for intrinsic volumes of Poisson k-flat processes
• Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs

Cited by

• Coherence of high-dimensional random matrices in a Gaussian case : application of the Chen-Stein method

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