Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-20T21:21:26.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbation analysis of functionals of random measures

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

François Baccelli ,
Maurice Klein  and
Sergei Zuyev
François Baccelli*
Affiliation:
INRIA, Sophia-Antipolis
Maurice Klein*
Affiliation:
CNET, Issy Les Moulineaux
Sergei Zuyev*
Affiliation:
INRIA, Sophia Antipolis
*
* Postal address: INRIA, Centre de Sophia Antipolis, 2004 Route des Lucioles, 06565 Valbonne Cedex, France.
** Postal address: CNET, 38 Rue du Général Leclerc, 92131 Issy les Moulineaux, France.
* Postal address: INRIA, Centre de Sophia Antipolis, 2004 Route des Lucioles, 06565 Valbonne Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

Keywords

STATIONARY RANDOM MEASURESSTATIONARY POINT PROCESSESPALM MEASURECAMPBELL MEASURELIGHT TRAFFIC ANALYSISPERTURBATION ANALYSISGRADIENT ESTIMATES IN COMPUTER SIMULATIONSSTOCHASTIC GEOMETRYVORONOI CELL

MSC classification

Primary: 60G55: Point processes
Secondary: 60G57: Random measures 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 306 - 325
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Mendendez Pelayo, Valencia, Spain, on 21–24 September 1993.

References

Baccelli, F. and Bremaud, P. (1993) Virtual customers in sensitivity and light traffic analysis via Campbell's formula for point processes. Adv. Appl. Prob. 25, 221–234.Google Scholar
Baccelli, F. and Bremaud, P. (1994) Elements of Queueing Theory. Springer-Verlag, New York.Google Scholar
Błaszczyszyn, B. (1993) Factorial moment expansion for stochastic systems. Preprint, Mathematical Institute, University of Wroclaw.Google Scholar
Borovkov, A. A. (1994) Asymptotic expansions for functionals of dilation of processes. Preprint, INRIA.Google Scholar
Choquet, G. (1964) Cours d'analyse II. Topologie. Masson, Paris.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
Dunford, N. and Schwartz, J. (1988) Linear Operators. Part I. Wiley, New York.Google Scholar
Ho, Yu-Chi and Cao, Xi-Ren (1991) Perturbation Analysis of Discrete Event Dynamic Systems. Kluwer, Boston, MA.CrossRefGoogle Scholar
Kerstan, J., Matthes, K. and Mecke, J. (1974) Unbegrenzt teilbare Punktprozesse. Akademie-Verlag, Berlin.Google Scholar
Kerstan, J., Matthes, K. and Mecke, J. (1982) Infinitely Divisible Point Processes. Nauka, Moscow (in Russian).Google Scholar
Mecke, J. (1967) Stationäre zufällige Masse auf lokal-kompakten abelschen Gruppe. Z. Wahrscheinlichkeitsth. 9, 36–58.Google Scholar
Møller, J. (1994) Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics 87, Springer-Verlag, New York.CrossRefGoogle Scholar
Møller, J. and Zuyev, S. A. (1995) Gamma-type results and other related properties of Poisson processes. Research report 282, Institute of Mathematics, University of Aarhus.Google Scholar
Neveu, J. (1976) Processus ponctuels. In Ecole d'Été de St. Flour, Lecture Notes in Mathematics 598, pp. 249–445. Springer-Verlag, Berlin.Google Scholar
Okabe, A., Boots, B. and Sugihara, K. (1992) Spatial Tesselations. Concepts and Applications of Voronoi Diagrams. Wiley, New York.Google Scholar
Reiman, M. and Simon, B. (1989) Open queueing systems in light traffic. Math. Operat. Res. 14, 26–59.Google Scholar
Russo, L. (1981) On the critical percolation probabilities. Z. Wahrscheinlichkeitsth. 56, 229–237.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar
Westcott, M. (1971) On existence and mixing results for cluster point processes. J. R. Statist. Soc. B 33, 290–300.Google Scholar
Zuyev, S. A. (1992) Estimates for distributions of the Voronoi polygon's geometric characteristics. Random Structures and Algorithms 3, 149–162.Google Scholar
Zuyev, S. A. (1993) Russo's formula for Poisson point fields and its applications. Discrete Math. Appl. 3, 63–73.Google Scholar