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Asymptotic Behaviour of an Empirical Nearest‐Neighbour Distance Function for Stationary Poisson Cluster Processes
Author(s) -
Heinrich Lothar
Publication year - 1988
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19881360109
Subject(s) - mathematics , combinatorics , central limit theorem , limit (mathematics) , order (exchange) , function (biology) , weak convergence , cluster (spacecraft) , nearest neighbour , mathematical analysis , statistics , computer security , finance , evolutionary biology , computer science , economics , asset (computer security) , biology , programming language , artificial intelligence
Summary. For stationary P OISSON cluster processes (PCP's) Ø on R 0 dthe limit behaviour, as v(D) → ∞, of the quantity \documentclass{article}\pagestyle{empty}\begin{document}$ \left({v\left(D \right)} \right)^{ - 1} \sum\limits_{x\varepsilon D:\phi \left({\left\{ x \right\}} \right) = 1} {\chi \left({x,r} \right)} $\end{document} , where χ( x, r ) = 1, if Ø( b ( x, r )) = 1, and χ( x, r ) = 0 otherwise, is studied. A central limit theorem for fixed r > 0 and the weak convergence of the normalized and centred empirical process on [0, R ] to a continuous G AUSS ian process are proved. Lower and upper bounds for the nearest neighbour distance function P 1 ({φ:Y(b(0,r))≧1}) of a stationary PCP are given. Moreover, a representation of higher order Palm distributions of PCP's and a central limit theorem for m ‐dependent random fields with unbounded m are obtained. Both these auxiliary results seems to be of own interest.