An Introduction to the Theory of Point Processes by GUJARATI

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The essence of this result is that, for a simple point process, the avoidance function alone is enough to determine the full set of fidi distributions. Our aim is to describe an interaction of structural properties of the space X and the function P0 (·) which are enough for P0 (·) to retain this determining character without the strong probabilistic assumptions of the Poisson process. Concerning terminology, Kendall (1974) used the term avoidance function in a more general (stochastic geometry) context, reflecting the fact that P0 (A) gives the probability of the support of a random set function avoiding a prescribed set A; other possible terms include zero function, avoidance probability function, and vacuity function [McMillan (1953)].

1). 21) for each fixed pair of bounded Borel sets. s. 22) for each fixed sequence of bounded Borel sets An with An ↓ ∅. 2. Finite-Dimensional Distributions and the Existence Theorem 29 an uncountable number of conditions to be checked, so that even though each individual condition is satisfied with probability 1, it cannot be concluded from this that the set on which they are simultaneously satisfied also has probability 1. XIV. s. for every Borel set A. But this implies that ξ ∗ and ξ have the same fidi distributions, and so completes the proof.

Over all possible partitions. But n E |ξ(Aj )| = E(Yn ) = j=1 and n µ(Aj ) j=1 1/2 ≥ n j=1 2 π 1/2 n µ(Aj ) , j=1 µ(Aj ) max1≤j≤n µ(Aj ) 1/2 1/2 = µ(A) max1≤j≤n µ(Aj ) 1/2 , so E(Yn ) can be made arbitrarily large by choosing a partition for which max1≤j≤n µ(Aj ) is sufficiently small. Because var Yn ≤ µ(A) for every partition, an application of Chebyshev’s inequality shows that for any given finite y, a partition can be found for which Pr{Yn ≥ y} can be made arbitrarily close to 1. s. bounded. Other examples may fail to be a random measure or point process because they fail to satisfy the bounded finiteness condition, as occurs for example with the jump points of many L´evy processes and for certain point sets for which Mandelbrot (1982, p.

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