Open Access
September 2005 Characterization of Palm measures via bijective point-shifts
Matthias Heveling, Günter Last
Ann. Probab. 33(5): 1698-1715 (September 2005). DOI: 10.1214/009117905000000224


The paper considers a stationary point process N in ℝd. A point-map picks a point of N in a measurable way. It is called bijective [Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York] if it is generating (by suitable shifts) a bijective mapping on N. Mecke [Math. Nachr. 65 (1975) 335–344] proved that the Palm measure of N is point-stationary in the sense that it is invariant under bijective point-shifts. Our main result identifies this property as being characteristic for Palm measures. This generalizes a fundamental classical result for point processes on the line (see, e.g., Theorem 11.4 in [Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York]) and solves a problem posed in [Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York] and [Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Ann. Inst. H. Poincaré Probab. Statist. 40 141–152]. Our second result guarantees the existence of bijective point-maps that have (almost surely with respect to the Palm measure of N) no fixed points. This answers another question asked by Thorisson. Our final result shows that there is a directed graph with vertex set N that is defined in a translation-invariant way and whose components are almost surely doubly infinite paths. This generalizes and complements one of the main results in [Holroyd, A. E. and Peres, Y. (2003). Electron. Comm. Probab. 8 17–27]. No additional assumptions (as ergodicity, nonlattice type conditions, or a finite intensity) are made in this paper.


Download Citation

Matthias Heveling. Günter Last. "Characterization of Palm measures via bijective point-shifts." Ann. Probab. 33 (5) 1698 - 1715, September 2005.


Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1111.60029
MathSciNet: MR2165576
Digital Object Identifier: 10.1214/009117905000000224

Primary: 60G55
Secondary: 60G10

Keywords: Palm measure , point process , point-map , point-stationarity , random graph , stationarity , tree

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 5 • September 2005
Back to Top