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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. 1) that will ensure that they are the fidi distributions of a random measure. The conditions fall into two groups: first the consistency requirements of the Kolmogorov existence theorem, and then the supplementary requirements of additivity and continuity needed to ensure that the realizations are measures.

Nk ) = Pk (Ai1 , . . , Aik ; ni1 , . . , nik ); ∞ (ii) r=0 Pk+1 (A1 , . . , Ak , Ak+1 ; n1 , . . , nk , r) = Pk (A1 , . . , Ak ; n1 , . . , nk ); (iii) for each disjoint pair of bounded Borel sets A1 , A2 , P3 (A1 , A2 , A1 ∪ A2 ; n1 , n2 , n3 ) has zero mass outside the set where n1 + n2 = n3 ; and (iv) for sequences {An } of bounded Borel sets with An ↓ ∅, P1 (An ; 0) → 1. IX, from which it follows that if the consistency conditions (i) and (ii) are satisfied for disjoint Borel sets, and if for such disjoint sets the equations n Pk (A1 , A2 , A3 , .

22 9. f. 9 (left- and right-hand figures respectively), locations reduced modulo 1 to the unit square. 1 By definition, a sequence of totally finite measures {µk } that converges weakly to a totally finite measure µ converges in the w# sense to the same limit. Check that the converse need not be true by taking µk to be Lebesgue measure on [0, k]. 2 (a) The sequence of measures {Nk } on BR defined by Nk = δ0 + δ1/k converges weakly to the measure N = 2δ0 ; each Nk ∈ NR#∗ but not the limit measure. (b) For k = 1, 2, .

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