 |
|
|
| | | | | |
|
|
|
|
|
Point shift characterization of Palm measures on Abelian groups
|
Matthias Heveling, Karlsruhe University
Gunter Last, Karlsruhe University
|
Abstract
Our first aim in this paper is to characterize Palm measures of
stationary point processes through point stationarity.
This generalizes earlier results from the Euclidean case
to the case of an Abelian group.
While a stationary point process looks statistically the
same from each site, a point stationary point process
looks statistically the same from each of its points.
Even in the Euclidean case our proof will simplify some of
the earlier arguments. A new technical result of some independent interest is the
existence of a complete countable family of matchings.
Using a change of measure we will generalize our results to discrete random
measures. In the Euclidean case we will finally treat general random measures
by means of a suitable approximation.
|
Full text: PDF
Pages: 122-137
Published on: February 4, 2007
|
Bibliography
- Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes.
Springer Series in Statistics. Springer-Verlag, New York, 1988. xxii+702 pp. ISBN: 0-387-96666-8 MR0950166 (90e:60060)
- Heveling, Matthias; Last, Günter. Characterization of Palm measures via bijective point-shifts.
Ann. Probab. 33 (2005), no. 5, 1698--1715. MR2165576
- Holroyd, Alexander E.; Peres, Yuval. Trees and matchings from point processes.
Electron. Comm. Probab. 8 (2003), 17--27 (electronic). MR1961286 (2004b:60127)
- Kallenberg, Olav. Foundations of modern probability.
Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2 MR1876169 (2002m:60002)
- Last, Günter. Stationary partitions and Palm probabilities.
Adv. in Appl. Probab. 38 (2006), no. 3, 602--620. MR2256871
- Matthes, Klaus. Stationäre zufällige Punktfolgen. I.
(German) Jber. Deutsch. Math.-Verein 66 1963/1964 Abt. 1 66--79. MR0160265 (28 #3479)
- Matthes, Klaus; Kerstan, Johannes; Mecke, Joseph. Infinitely divisible point processes.
John Wiley & Sons, Chichester-New York-Brisbane, 1978. xii+532 pp. ISBN: 0-471-99460-X MR0517931 (58 #24538)
- Mecke, J. Stationäre zufällige Masse auf lokalkompakten Abelschen
(German) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 1967 36--58. MR0228027 (37 #3611)
- Mecke, J. Invarianzeigenschaften allgemeiner Palmscher Maße.
(German) Math. Nachr. 65 (1975), 335--344. MR0374385 (51 #10585)
- Neveu, J. Processus ponctuels.
pp. 249--445. Lecture Notes in Math., Vol. 598, Springer-Verlag, Berlin, 1977. MR0474493 (57 #14132)
- Thorisson, Hermann. Coupling, stationarity, and regeneration.
Probability and its Applications (New York). Springer-Verlag, New York, 2000. xiv+517 pp. ISBN: 0-387-98779-7 MR1741181 (2001b:60003)
- Timár, Ádám. Tree and grid factors for general point processes.
Electron. Comm. Probab. 9 (2004), 53--59 (electronic). MR2081459 (2005h:60145)
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context