Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Eléments de mathématique. XIII. Première partie: Les structures fondamentales de l’analyse. Livre VI: Intégration. Chapitre I: Inégalités de convexité. Chapitre II: Espaces de Riesz. Chapitre III: Mesures sur les espaces localement compacts. Chapitre IV: Prolongement d’une mesure; espaces 𝐿^{𝑝}, Actualités Sci. Ind., no. 1175, Hermann et Cie, Paris, 1952 (French). MR 0054691
  • [2] Werner Fieger, Die Anzahl der 𝛾-Niveau-Kreuzungspunkte von stochastischen Prozessen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 227–260 (German). MR 0290437
  • [3] Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
  • [4] K. Murali Rao, On decomposition theorems of Meyer, Math. Scand. 24 (1969), 66–78. MR 0275510
  • [5] F. Papangelou, On the Palm probabilities of processes of points and processes of lines, Stochastic geometry (a tribute to the memory of Rollo Davidson), Wiley, London, 1974, pp. 114–147. MR 0402917
  • [6] -, Summary of some results on point and line processes, Proc. IBM Conference on Stochastic Point Processes (held August 1971).
  • [7] Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR 0056821
  • [8] Czesław Ryll-Nardzewski, Remarks on processes of calls, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 455–465. MR 0140153
  • [9] František Zítek, The theory of ordinary streams, Select. Transl. Math. Statist. and Probability, Vol. 2, American Mathematical Society, Providence, R.I., 1962, pp. 241–251. MR 0150827

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60G10, 60K99

Retrieve articles in all journals with MSC: 60G10, 60K99

Additional Information

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