1 results
The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process
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- Journal:
- Journal of Applied Probability / Volume 15 / Issue 2 / June 1978
- Published online by Cambridge University Press:
- 14 July 2016, pp. 433-439
- Print publication:
- June 1978
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- Article
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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.
where 
and 
for 0 < 