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

✍ Scribed by Lothar Heinrich

• Publisher: John Wiley and Sons
• Year: 1988
• Tongue: English
• Weight: 789 KB
• Volume: 136
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19881360109

✦ Synopsis

For stationary POISWR duster processes (PCP's) @ on Rd the limit behirviour, as v(D) --f 00, of the quantity C xfz. r), whem ~(z, r) = ly if @(b(z,r)) = 1, and ~(z, r ) = 0 otherwise, is studied. A mid ther>rem for fired r > 0 snd the weak convergence of the normdkd snd e e d eBqkkxb1 procefs on [O, B] to a wntinuona QAmSian process are provd Lower end upper bounds for the neweat neighboar diatance function Pi((cp: (p(b(0, r)) 5 11) of a stationary PCP are byen. MoreOoer, B representotian of higher order Palm distributiona of WE and s central iimit t h r e m for m-dependent random fields with unbounded m are obtsined. Both these auxiliary resnlta Beems e0 be of own intierest. ZED: *c(xl,-t An important quantity for the st&istical mdy& of shtiionq point processes is the distribution function D(T) of the distanoe to the n-t neighborn of a given point of the point process under consideration (see [S, 191). In the present paper we are concerned throughout with (ststionary) Poissos cluster processes. "he paper is organized as folxom. Subsequently, in this section we give a series of definitions and sketch the theoretical background. A detded study of the theory point processes one can find in [16, 191 (in prticular for PCP's see [I, 2,233) and for the statistical analysis of point processes the reader is referred to [7, 17, 211. In 8ed. 2 we formulate and prove a representation formula for the k-th order pat^ diatribution of PCPs. Noh that this d t is independent of the rempart of the peper. Section 3 contains 8 LwDasma-type theorem for %-dependent random fields with mbonndedly growing m which is a key to prove the main results in Sect. 5. In Sect. 4 we derive lower and upper bounds of the nearest neighbour distance function ("DF) Dfr). Finally, in Sect. 5 we prove a central limit theorem (CLT) and a corresponding fnnctiond limit theorem for 8n unbiased estimator of 1 -D(r) construated from an obeerrvrrtion of the PCP on a finite region W (sampling windoff) whose area tends to infinity. Let dl be the set of d l l d y finite ~ounting mewtp on the d-dimensionad Evcmean space Rd (equipped with it8 a-algebra of BOBEL seta 8 6 ) and let Cm be the 9* 1 32 Math. Nachr. 136 (1988)