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PR ] 2 9 M ar 2 01 8 Multivariate second order Poincaré inequalities for Poisson functionals
| Content Provider | Semantic Scholar |
|---|---|
| Author | Schulte, Matthias Yukich, J. E. |
| Copyright Year | 2018 |
| Abstract | Given a vector F = (F1, . . . , Fm) of Poisson functionals F1, . . . , Fm, we establish quantitative bounds for the proximity between F and an m-dimensional centered Gaussian random vector NΣ with covariance matrix Σ ∈ Rm×m. We derive results for the d2and d3-distances based on smooth test functions as well as for the dconvex-distance and the dHl-distance given by dHl(F,NΣ) := sup h∈Hl |Eh(F )− Eh(NΣ)|, a multi-dimensional generalization of the Kolmogorov distance, where l ∈ N and Hl is the set of indicator functions of intersections of l closed half-spaces in Rm. The bounds are multivariate counterparts of the second order Poincaré inequalities of [15] and, as such, are expressed in terms of integrated moments of first and second order difference operators. The derived second order Poincaré inequalities for nonsmooth test functions, which are of the same order as for smooth test functions, are made possible by new bounds on the derivatives of solutions to the Stein equation for the multivariate normal distribution, which might be of independent interest. We present applications to the multivariate normal approximation of first order Wiener-Itô integrals and of statistics of Boolean models. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://export.arxiv.org/pdf/1803.11059 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
