REMARK ON THE PALM INTENSITY OF
NEYMAN-SCOTT CLUSTER POINT PROCESSES
Toshima-ku, Tokyo 171-8501, JAPAN
Dedicated to Professor Yosihiko Ogata on
his 66th birthday with respect and affection.
Abstract. The present paper is concerned with notes on the Palm intensity of
Neyman-Scott cluster point processes. The Palm intensity is known as
the most important and informative second-order characteristic that
plays a role of a configurational criterion for point patterns. It
is remarkable that its pole at the origin and range of correlation
play some intrinsic roles in the point pattern analysis. However,
their generic forms cannot be derived due to the inability to
explicitly describe the Palm intensity except in typical cluster
point processes. This paper provides a sufficient condition for the
existence of the pole and a bound on the range of correlation for
more general cluster point processes. As applications of our
results, we give an exact formula for a variance-area curve of the
Neyman-Scott cluster point processes and a remark on a sufficient
condition for a Neyman-Scott cluster point process with uniformly
bounded diameter to be a connected component Markov point process.
The current results play a key role in the identification problem on
a superposed Neyman-Scott cluster point process model due to the
Palm intensity, which is the most significant pragmatic contribution
made in this paper.
AMS Subject classification: 60G55, 60D05
Keywords and phrases: Neyman-Scott cluster point process, Palm intensity, pole, range of correlation, identification problem
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