Characterization of Palm measures via bijective point-shifts
Characterization of Palm measures via bijective point-shifts
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通过双射点位移表征棕榈测量
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发表时间：
2005
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通讯作者：
Matthias Heveling
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作者：
Matthias Heveling
本文考虑Rd中的一个平稳点过程N.点映射以可测量的方式选取N中的一个点。它被称为双射[Thorisson，H.（2000年）的第10/2002号决议。耦合、稳定性和再生。Springer，纽约]如果它（通过适当的移位）生成N上的一个双射映射。Mecke [Math. Nachr. 65（1975）335-344]证明了N的Palm测度是点平稳的，因为它在双射点位移下是不变的。我们的主要结果确定此属性为Palm措施的特点。这推广了直线上点过程的一个基本经典结果（参见，例如，定理11.4 [Kallenberg，O.（2002年）的报告。Foundations of Modern Probability，2nd ed. Springer，纽约]）并解决了[Thorisson，H.（2000年）的第10/2000号决议。耦合、稳定性和再生。Springer，纽约]和[Ferrari，P.A.，兰丁角和Thorisson，H.（2004年）。安·因斯特·H庞加莱概率中央集权主义者40 141-152]。我们的第二个结果保证了双射点映射的存在，这些点映射（对于N的Palm测度几乎肯定）没有不动点。这回答了Thorisson提出的另一个问题。我们的最终结果表明，存在一个顶点集为N的有向图，它是以一种交换不变的方式定义的，并且它的分支几乎必然是双无限路。这推广和补充了[Holroyd，A. E.和Peres，Y.（2003年）的报告。电子通讯公司8 17-27]。没有额外的假设（如遍历性，非格型条件，或有限强度），本文。
The paper considers a stationary point process N in R d . 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. Poincare 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. Prohab. 8 17-27]. No additional assumptions (as ergodicity, nonlattice type conditions, or a finite intensity) are made in this paper.