Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

By Günter Last, Mathew D. Penrose, Matthias Schulte, C. Thaele. Advances in Applied Probability, 2012

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This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in .

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/05cfec382a46cc04b61f29a6a7d4efe4faa1ec44

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In Applied Mathematics

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Citations in the corpus

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “e CLTs in a geometric setting. Other relevant contributions in this area are: [16, 49] for explicit bounds in the Kolmogorov distance; [18] for applications to additive functionals of Boolean models; [7, 14, 23, 24, 28] for applications to geometric U-statistics and other non-linear functionals; [50”
Stochastic analysis for Poisson processes (2014, background): “We again refer the reader to [29] as well as to [22, 15].”
New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, methodology): “In particular, we consider a much wilder class of geometric functionals, which is inspired by the findings in [14].”
Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, unknown): “Z Ii−1(f(z,·))Ij−1(g(z,·))λ(dz) 2 = ZZ EIi−1(f(y,·))Ii−1(f(z,·))Ij−1(g(y,·))Ij−1(g(z,·))λ(dy)λ(dz), and E Z Ii−1(f(z,·))Ij−1(g(z,·))λ(dz) = ( (i−1)!hf,gii, i= j, 0, i6= j, (8.11) with Theorem 3.1 in [16] (see also [24, Corollary 7.2] and [32, Proposition 3.1]) proves the ﬁrst equation. The second id”