Normal Approximation of Poisson Functionals in Kolmogorov Distance
By Matthias Schulte. Journal of theoretical probability, 2012
55 citations recorded by the source corpus.
Peccati, Solè, Taqqu, and Utzet recently combined Stein’s method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper was to show this behavior for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.
Subfields: Applied Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/dea5876cb3ec2116e0522f58cdf58fec1a578cce
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In Applied Mathematics
- Matthias Schulte
- Günter Last
- Giovanni Peccati
- Raphaël Lachièze-Rey
- C. Thaele
- Peter Eichelsbacher
- Joseph E. Yukich
- Mathew D. Penrose
In Statistics and Probability
- Matthias Schulte
- Günter Last
- Giovanni Peccati
- C. Thaele
- Raphaël Lachièze-Rey
- Peter Eichelsbacher
- Joseph E. Yukich
- Mathew D. Penrose
Citations in the corpus
- Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “3 in [49] the assertion is stated for g ∈ domD with finite chaos expansions.”
- Normal approximation for stabilizing functionals (2017, methodology): “Bounds for the normal approximation of such random variables were deduced, for example, in [35] and [21] for the Wasserstein distance and in [43] and [14] for the Kolmogorov distance.”
- New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, methodology): “• We shall compare our bound with that for dK(F,Z) from [36].”