Get An Introduction to the Theory of Point Processes, Volume II: PDF

By D.J. Daley; David Vere-Jones

ISBN-10: 0387213376

ISBN-13: 9780387213378

ISBN-10: 0387498354

ISBN-13: 9780387498355

This can be the second one quantity of the remodeled moment variation of a key paintings on element method concept. absolutely revised and up to date by way of the authors who've transformed their 1988 first variation, it brings jointly the fundamental concept of random measures and element approaches in a unified atmosphere and keeps with the extra theoretical themes of the 1st variation: restrict theorems, ergodic idea, Palm conception, and evolutionary behaviour through martingales and conditional depth. The very huge new fabric during this moment quantity contains accelerated discussions of marked element methods, convergence to equilibrium, and the constitution of spatial element approaches.

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Extra resources for An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, 2nd Edition

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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 , .

Concerning the existence of point processes, two basic approaches are widely used in the literature. A point process is defined sometimes as an integer-valued random measure N (·, ω), as above, and sometimes as a sequence of random variables {yi }. When are these approaches equivalent? In one direction the argument is straightforward and covered in the next result where we start from certain finite or countably infinite sequences {yi : i = 1, 2, . } of X -valued random elements. X. Let {yi } be a sequence of X -valued random elements defined on a probability space (Ω, E, P), and suppose that there exists an / E0 implies that for any bounded event E0 ∈ E such that P(E0 ) = 0 and ω ∈ set A ∈ BX , only a finite number of the elements of {yi (ω)} lie within A.

Aik ; xi1 , . . , 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, Volume II: General Theory and Structure, 2nd Edition by D.J. Daley; David Vere-Jones


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