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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integrability of expected increments of point processes and a related random change of scale

Author: F. Papangelou
Journal: Trans. Amer. Math. Soc. 165 (1972), 483-506
MSC: Primary 60G10; Secondary 60K99
MathSciNet review: 0314102
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Abstract: Given a stationary point process with finite intensity on the real line R, denote by $ N(Q)$ (Q Borel set in R) the random number of points that the process throws in Q and by $ {\mathcal{F}_t}(t \in R)$ the $ \sigma $-field of events that happen in $ ( - \infty ,t)$. The main results are the following. If for each partition $ \Delta = \{ b = {\xi _0} < {\xi _1} < \cdots < {\xi _{n + 1}} = c\} $ of an interval [b, c] we set $ {S_\Delta }(\omega ) = \sum\nolimits_{\nu = 0}^n {E(N[{\xi _\nu },{\xi _{\nu + 1}})\vert{\mathcal{F}_{{\xi _\nu }}})} $ then $ {\lim _\Delta }{S_\Delta }(\omega ) = W(\omega ,[b,c))$ exists a.s. and in the mean when $ {\max _{0 \leqq \nu \leqq n}}({\xi _{\nu + 1}} - {\xi _\nu }) \to 0$ (the a.s. convergence requires a judicious choice of versions). If the random transformation $ t \Rightarrow W(\omega ,[0,1))$ of $ [0,\infty )$ onto itself is a.s. continuous (i.e. without jumps), then it transforms the nonnegative points of the process into a Poisson process with rate 1 and independent of $ {\mathcal{F}_0}$. The ratio $ {\varepsilon ^{ - 1}}E(N[0,\varepsilon )\vert{\mathcal{F}_0})$ converges a.s. as $ \varepsilon \downarrow 0$. A necessary and sufficient condition for its convergence in the mean (as well as for the a.s. absolute continuity of the function $ W[0,t)$ on $ (0,\infty ))$ is the absolute continuity of the Palm conditional probability $ {P_0}$ relative to the absolute probability P on the $ \sigma $-field $ {\mathcal{F}_0}$. Further results are described in §1.

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Keywords: Stochastic point process, Poisson process, Palm probability, Burkill integrability, conditional expectation, almost sure convergence, mean convergence, absolute continuity
Article copyright: © Copyright 1972 American Mathematical Society