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
DOI: https://doi.org/10.1090/S0002-9947-1972-0314102-9
MathSciNet review: 0314102
Full-text PDF

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, Livre VI: Intégration, Chap. 4, Actualités Sci. Indust., no. 1175, Hermann, Paris, 1952. MR 14, 960. MR 0054691 (14:960h)
  • [2] W. Fieger, Die Anzahl der $ \gamma $-Niveau-Kreuzungspunkte von stochastischen Prozessen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 227-260. MR 0290437 (44:7618)
  • [3] P. A. Meyer, Probability and potentials, Publ. Inst. Math. Univ. Strasbourg, no. 14, Actualités Sci. Indust., no. 1318, Hermann, Paris, 1966; English transl., Blaisdell, Waltham, Mass., 1966. MR 34 #5118; MR 34 #5119. MR 0205288 (34:5119)
  • [4] K. Murali Rao, On decomposition theorems of Meyer, Math. Scand. 24 (1969), 66-78. MR 0275510 (43:1264)
  • [5] F. Papangelou, On the Palm probabilities of processes of points and processes of lines, Memorial Volume to R. Davidson (edited by D. G. Kendall) (to appear). MR 0402917 (53:6731)
  • [6] -, Summary of some results on point and line processes, Proc. IBM Conference on Stochastic Point Processes (held August 1971).
  • [7] F. Riesz and B. Sz.-Nagy, Leçons d'analyse fonctionnelle, 2nd ed., Akad. Kiadó, Budapest, 1953; English transl., Ungar, New York, 1955. MR 17, 175. MR 0056821 (15:132d)
  • [8] C. Ryll-Nardzewski, Remarks on processes of calls, Proc. Fourth Berkeley Sympos. Math. Statist, and Prob., vol. 2, Univ. California Press, Berkeley, Calif., 1961, pp. 455-465. MR 25 #3575. MR 0140153 (25:3575)
  • [9] F. Zítek, The theory of ordinary streams, Czechoslovak Math. J. 8 (83) (1958), 448-459; English transl., Selected Transl. Math. Statist, and Prob., vol. 2, Amer. Math. Soc., Providence, R. I., 1962, pp. 241-251. MR 21 #376; MR 27 #814. MR 0150827 (27:814)

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

DOI: https://doi.org/10.1090/S0002-9947-1972-0314102-9
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

American Mathematical Society