March 28, 2017

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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. Proof. X. 17) when written in the form r P{ζn (A) = k} = lim P{N (A) ≤ r} = 1 lim lim r→∞ n→∞ k=0 r→∞ and expresses the fact that a point process N is boundedly finite. For the sufficiency, it is clear from (i) and (ii) that we can construct an indicator process Z on bounded A ∈ R with fidi distributions (for any finite number k of disjoint bounded A1 , .

Conclude that any MPP is equivalent to another MPP whose ground process is simple. 7 Let {X(t): t ∈ R} be a measurable nonnegative stochastic process on (Ω, E, P). Show that, when the integrals concerned are finite, the relation X(t, ω) dt ξ(A, ω) = (bounded A ∈ BR ) A defines a random measure ξ: Ω → M# R. 8 Let N be a well-defined point process on X = R2 . With each point yi in a realization of N associate a geometric object in one of the following ways. (a) Construct a disk Sr (yi ) with centre yi and radius r, and let ξ(A) = È i (A ∩ Sr (yi )) (bounded A ∈ B(R2 )) represent the total area of disks intersecting any Borel set A.

III asserts that finite families of disjoint sets from a semiring A generating BX form a determining class for random measures on (X , BX ). Of course we also have the following corollary. IV. The distribution of a random measure is completely determined by its fidi distributions. 9), namely, for bounded Borel sets A1 , A2 , . . and nonnegative integers n1 , n2 , . . , Pk (A1 , . . , Ak ; n1 , . . , nk ) = P{N (Ai ) = ni (i = 1, . . , k)}. VIII, the distribution of a point process, meaning the measure induced on NX# , B(NX# ) , is completely specified by the fidi distributions of N (A) for A in a countable ring generating the Borel sets.

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