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Poisson limits and nonparametric estimation for pairwise interaction point processes

Part of: Nonparametric inference Inference from stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Wei-Bin Chang  and
John A. Gubner
Wei-Bin Chang*
Affiliation:
University of Wisconsin-Madison
John A. Gubner*
Affiliation:
University of Wisconsin-Madison
*
∗Postal address: 58-1 Chung Shan Rd, Cholan, Miaoli 36901, Taiwan. Email address: [email protected]
∗∗Postal address: Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA. Email address: [email protected]
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Abstract

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Keywords

Pairwise interaction point processinteraction functionnonparametric estimation

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62G07: Density estimation 62M30: Spatial processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 252 - 260
Copyright
Copyright © 2000 by The Applied Probability Trust 

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Footnotes

This work was supported by the Office of Naval Research, Mathematical Sciences Division, under ONR Grant N00014–94–1–0366

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