Multiple-point hit distribution functions and vague convergence of related measures

✍ Scribed by Felix Ballani

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 138 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200810287

✦ Synopsis

Abstract

For a stationary and isotropic random closed set Z in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^d$\end{document} it is a well‐known fact that its covariance C(t) and its spherical contact distribution function \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widetilde{H}_s(t)$\end{document} admit at t = 0 a derivative which is a multiple of the surface intensity of Z. Within the quite general setting of gentle sets, Kiderlen and Rataj 10 show a more general result (covering both previous cases) for the derivative of a hit distribution function of Z with respect to a structuring element which only needs to be compact and contains the origin. Using this general setting the present paper introduces m‐point hit distribution functions of Z, m ⩾ 2, and shows how they are related to the __m__th‐order surface product density of Z. This also generalizes a result of Ballani 1 for the two‐point spherical contact distribution function of a germ‐grain model. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim