Asymptotic Behaviour of an Empirical Nearest‐Neighbour Distance Function for Stationary Poisson Cluster Processes

Abstract

Abstract Summary. For stationary P OISSON cluster processes (PCP's) Ø on R the 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.