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The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process

Published online by Cambridge University Press:  14 July 2016

A. M. Liebetrau
A. M. Liebetrau*
Affiliation:
The Johns Hopkins University
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Abstract

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

Keywords

WEAK CONVERGENCEVARIANCE FUNCTIONTWO-DIMENSIONAL POISSON PROCESSMULTIDIMENSIONAL POISSON PROCESSESTIMATOR
Type
Short Communications
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 433 - 439
Copyright
Copyright © Applied Probability Trust 1978 

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