Characterization of palm measures via bijective point-shifts

• 成果类型: Article
• 署名作者: Heveling, M; Last, G
• 署名单位: Helmholtz Association; Karlsruhe Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000224
• 发表日期: 2005
• 页码: 1698-1715
• 关键词: trees

摘要

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 pointstationary 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. Poincan 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.