Palm measure duality and conditioning in regenerative sets. (English) Zbl 0953.60037

Kallenberg, Olav

Ann. Probab. 27, No. 2, 945-969 (1999).

For a simple point process \(\Xi\) on a suitable topological space, the associated Palm distribution at a point \(s\) is a limit of the conditional distribution given that \(\Xi\) intersects a neighborhood \(I\) of \(s\), as \(I\) shrinks to \(s\). The author studies the corresponding approximation problem for more general random sets \(\Xi\), which, in particular, do not need to be locally finite. For this purpose he shows that the corresponding Palm distributions can be expressed in terms of certain conditional densities. These possess martingale properties which allow to treat the approximation problem successfully as a limiting problem involving conditional hitting probabilities. As an extensive application of this dual approach, Palm distributions of regenerative sets with respect to their local time random measures are studied.

Reviewer: R.Schassberger (Braunschweig)

MSC:

60G57

Random measures

Keywords:

random measures; Palm distributions; conditional hitting probabilities; local time; regenerative sets

References:

[1]	Assouad, P. (1971). Démonstration de la ”Conjecture de Chung” par Carleson. Séminaire de Probabilités V. Lecture Notes in Math. 191 17-20. Springer, Berlin.
[2]	Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York. · Zbl 0169.49204
[3]	Bretagnolle, J. (1971). Résultats de Kesten sur les processus a accroissements indépendants. Séminaire de Probabilités V. Lecture Notes in Math. 191 21-36. Springer, Berlin.
[4]	Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. · Zbl 0657.60069
[5]	Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1987, 1992). Probabilités et Potentiel. Hermann, Paris.
[6]	Dellacherie, C. and Meyer, P. A. (1980). In Probabilités et Potential. Hermann, Paris.
[7]	Ivanoff, B. G., Merzbach, E. and Schiopu-Kratina, I. (1993). Predictability and stopping on lattices of sets. Probab. Theory Related Fields 97 433-446. · Zbl 0794.60025
[8]	Kallenberg, O. (1981). Splitting at backward times in regenerative sets. Ann. Probab. 9 781- 799. · Zbl 0526.60061
[9]	Kallenberg, O. (1982). Characterizations and embedding properties in exchangeability.Wahrsch. Verw. Gebiete 60 249-281. · Zbl 0481.60019
[10]	Kallenberg, O. (1986). Random Measures, 4th ed. Academic Press, New York. · Zbl 0345.60032
[11]	Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York. · Zbl 0892.60001
[12]	Krickeberg, K. (1956). Convergence of martingales with a directed index set. Trans. Amer. Math. Soc. 83 313-337. · Zbl 0083.27501
[13]	Kurtz, T.G. (1980). The optional sampling theorem for martingales indexed by directed sets. Ann. Probab. 8 675-681. · Zbl 0442.60045
[14]	Maisonneuve, B. (1974). Syst emes régénératifs. Astérisque 15. · Zbl 0285.60049
[15]	Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester. · Zbl 0383.60001