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ISBN-13: 9780072427929

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Extra resources for An Introduction to the Theory of Point Processes

Example text

Hence the ti (N ), and also therefore the τi (N ), are random variables whenever N is a simple point process. 19) implies that the resulting point process is boundedly finite. X that N is a well-defined, simple point process. 14 provides a further example of this type and a counterexample involving the subset N0 ⊂ NR#∗ with an atom at 0 and the subset S0+ for which t0 = x = 0). Sigman’s (1995) Appendix D discusses these questions also. There is no analogous simple representation for point processes in Rd , d = 2, 3, .

Tn−1 , tn , . . When applicable, 0 satisfies t−1 < 0 ≤ t0 , so that the interval of length τ0 contains 0. 18) defines a mapping R: NR#∗ → S + . The inverse mapping R−1 is defined for s+ ∈ S + by t0 (s+ ) = −x0 (s+ ), ti (s+ ) = ti−1 (s+ ) + τi ti+1 (s+ ) − τi+1 (i ≥ 1), (i < 0). 20) We give S + the Borel σ-algebra B(S + ) obtained in the usual way as the product of σ-algebras on each copy of R+ . XII. 18), R say, provides a one-to-one both ways measurable mapping of NR#∗ into S + . In particular, (i) the quantities τi (N ) and x(N ) are well-defined random variables when N is a simple point process; and (ii) there is a one-to-one correspondence between the probability distributions P ∗ of simple point processes on NR#∗ and probability distributions on the space S + .

Xik ). (b) Consistency of marginals. For all k ≥ 1, Fk+1 (A1 , . . , Ak , Ak+1 ; x1 , . . , xk , ∞) = Fk (A1 , . . , Ak ; x1 , . . , xk ). The first of these conditions is a notational requirement: it reflects the fact that the quantity Fk (A1 , . . , Ak ; x1 , . . , xk ) measures the probability of an event {ω: ξ(Ai ) ≤ xi (i = 1, . . , k)}, that is independent of the order in which the random variables are written down. The second embodies an essential requirement: it must be satisfied if there is to exist a single probability space Ω on which the random variables can be jointly defined.

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An Introduction to the Theory of Point Processes by GUJARATI

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