By GUJARATI

ISBN-10: 0072427922

ISBN-13: 9780072427929

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**Extra resources for An Introduction to the Theory of Point Processes**

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22), and because the number of sets A ∈ A and integers k < ∞ is countable, this requirement is satisﬁed almost surely. s. extended to a measure on σ(A). The necessity of both conditions follows directly from the additivity and continuity properties of a measure. s. 24b) 18 9. XV. s. ﬁnite-valued (ﬁnite integer-valued) on bounded Borel sets. 24b)] hold for all sequences {An } of bounded Borel sets with An ↓ ∅. Proof. IX. 22) hold for Borel sets in general, they certainly hold for sets in A. XIV are satisﬁed, and we can assert that with probability 1 the ξA (ω), initially deﬁned for A ∈ A, can be extended to measures ξ ∗ (A, ω) deﬁned for all A ∈ σ(A) = BX .

X , meaning a countable family T = {Tn } of ‘inﬁnite’ dissecting systems of 16 9. 3 Tilings Tn = {Ani }, Tn+1 = {An+1,i } with sets containing x, y. X : each tiling Tn = {Ani : i = 1, 2, . } consists of countably many disjoint bounded sets Ani satisfying (i) (Partition and tiling properties) Ani ∩Anj = ∅ for i = j, and i Ani = X ; (ii) (Nesting property) An−1,i ∩ Anj = Anj or ∅, and any An−1,i ∈ Tn−1 is expressible An−1,i = j∈Jn−1,i Anj for some ﬁnite set of indices Jn−1,i ; (iii) (Point-separating property) given distinct x, y ∈ X , there exists an inte/ Ani ; and ger n(x, y) such that for n ≥ n(x, y), x ∈ Ani implies y ∈ (iv) (Enumeration consistency property) given distinct x, y ∈ X , when Tm is such that x ∈ Ami y, for all n ≥ m sets An,in x, An+1,in+1 x and An,jn y, An+1,jn+1 y are such that jn − in and jn+1 − in+1 have the same sign.

VI are not only necessary but also suﬃcient to ensure that the ﬁdi distributions can be associated with a random measure. 1(f) implies that without nonnegativity, the suﬃciency argument breaks down. It appears to be an open problem to ﬁnd necessary and suﬃcient conditions on the ﬁdi distributions that ensure that they belong to a random signed measure. 4. VII. V. VI. Proof. XIV, so we proceed to suﬃciency. 1). 21) for each ﬁxed pair of bounded Borel sets. s. 22) for each ﬁxed sequence of bounded Borel sets An with An ↓ ∅.

### An Introduction to the Theory of Point Processes by GUJARATI

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