Doubly Stochastic Poisson Processes by J. Grandell

By J. Grandell

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Extra resources for Doubly Stochastic Poisson Processes

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We will now give for doubly stochastic Poisson processes.

0,I~ such that lim Pn = 0 and P]'P2 .... ~ P be given. n-~ Then Dpn Pn w some P0~P if and only if Apn Pn w some H0~ ~ and in this case P0 = PH 0' Proof The proof is given by Kallenberg (1975:1) and will not be reproduced here. 9 22 Theorem 5 generalizes earlier limit results for p-thinnings. As noted by Kallenberg also Mecke~s characterization of doubly stochastic Poisson processes is a simple consequence. To see that, let P0 ~ P n ~ D P be given. e. PO ~ D. Further Apn p n w An P n ~ is the same as Ap D~IpH n n 0 w H0 HO' which is a result due to Kerstan, Matthes and Mecke (1974, p 315), from which theorem I follows.

Led by the elementary definition of condi- tional probabilities we consider the ratio Pr{N6 Bg~ B(x) } Pr{NEB(x)} . e. x,(N ) " . e. multiple pointsE ~IN)N{dx}d~ not occtu-, we have IB(x)(N) = N[dx}. Thus we are led to consider E N(dx} and this will be our Palm probability. It will turn out, see theorem 2, 54 that it is convenient for our purposes to use this definition of PaLm probabilities also for point processes which are not simple and for general random measures. For a random measure A, we by formal analogy define a Palm E IB(A)A{dx} p r ~ a b i l i t y by E A{dx} We conclude this heuristic reasoning by some remarks on the definition used by Jagers (1973).

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