Integrability of expected increments of point processes and a related random change of scale
HTML articles powered by AMS MathViewer
- by F. Papangelou
- Trans. Amer. Math. Soc. 165 (1972), 483-506
- DOI: https://doi.org/10.1090/S0002-9947-1972-0314102-9
- PDF | Request permission
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 ${S_\Delta }(\omega ) = \sum \nolimits _{\nu = 0}^n {E(N[{\xi _\nu },{\xi _{\nu + 1}})|{\mathcal {F}_{{\xi _\nu }}})}$ 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 ${\varepsilon ^{ - 1}}E(N[0,\varepsilon )|{\mathcal {F}_0})$ 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
- 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 Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann & Cie, Paris, 1952 (French). MR 0054691
- Werner Fieger, Die Anzahl der $\gamma$-Niveau-Kreuzungspunkte von stochastischen Prozessen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 227–260 (German). MR 290437, DOI 10.1007/BF00563139
- Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
- K. Murali Rao, On decomposition theorems of Meyer, Math. Scand. 24 (1969), 66–78. MR 275510, DOI 10.7146/math.scand.a-10920
- 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 —, Summary of some results on point and line processes, Proc. IBM Conference on Stochastic Point Processes (held August 1971).
- Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR 0056821
- 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
- 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
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- 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