An Introduction to the Theory of Point Processes, 2nd by GUJARATI

By GUJARATI

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Ak }, z Fk (A1 , A2 , A3 , . . , Ak ; dx1 , z − x1 , x3 , . . 7) = Fk−1 (A1 ∪ A2 , A3 , . . , Ak ; z, x3 , . . , xk ). Proof. VI(a) and therefore necessary. We show that it is also sufficient. Let us first point out how the extension from disjoint to arbitrary families of sets can be made. Let {B1 , . . , Bn } be any such arbitrary family. Then there exists a minimal family {A1 , . . , Ak } of disjoint sets (formed from the nonempty intersections of the Bi and Bic ) such that each Bi can be represented as a finite union of some of the Aj .

22), which express the fact that the random variables so produced must fit together as measures. VI (Measure Requirements). (a) Additivity. For every pair A1 , A2 of disjoint Borel sets from BX , the distribution F3 (A1 , A2 , A1 ∪A2 ; x1 , x2 , x3 ) is concentrated on the diagonal x1 + x2 = x3 . (b) Continuity. For every sequence {An : n ≥ 1} of bounded Borel sets decreasing to ∅, and all > 0, 1 − F1 (An ; ) → 0 (n → ∞). V imply the existence of a probability space on which the random variables ξ(A), A ∈ BX , can be jointly defined.

XV (Kurtz, 1974). s. X . In order that there exist a point process on X with avoidance function ψ, it is necessary and sufficient that (i) ψ be completely monotone; (ii) ψ(∅) = 1; (iii) ψ(An ) → 1 for any bounded sequence {An } in R for which An → ∅ (n → ∞); and (iv) for every bounded A ∈ R, r lim lim r→∞ n→∞ k ∆(Ani1 , . . ,ir } Anij = 1, j=1 where {Tn } = {Ani : i = 1, . . , kn } is a dissecting system for A, {Tn } ⊆ R, and the inner summation is over all distinct combinations of k sets from the kn sets in the partition Tn for A.

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