The Annals of Probability

Characterization of Palm measures via bijective point-shifts

Matthias Heveling and Günter Last

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Abstract

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.

Article information

Source
Ann. Probab., Volume 33, Number 5 (2005), 1698-1715.

Dates
First available in Project Euclid: 22 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1127395870

Digital Object Identifier
doi:10.1214/009117905000000224

Mathematical Reviews number (MathSciNet)
MR2165576

Zentralblatt MATH identifier
1111.60029

Subjects
Primary: 60G55: Point processes
Secondary: 60G10: Stationary processes

Keywords
Point process stationarity Palm measure point-map random graph tree point-stationarity

Citation

Heveling, Matthias; Last, Günter. Characterization of Palm measures via bijective point-shifts. Ann. Probab. 33 (2005), no. 5, 1698--1715. doi:10.1214/009117905000000224. https://projecteuclid.org/euclid.aop/1127395870


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