The Annals of Probability

Invariant transports of stationary random measures and mass-stationarity

Günter Last and Hermann Thorisson

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We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu’s exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary.

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Ann. Probab., Volume 37, Number 2 (2009), 790-813.

First available in Project Euclid: 30 April 2009

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Zentralblatt MATH identifier

Primary: 60G57: Random measures 60G55: Point processes
Secondary: 60G60: Random fields

Stationary random measure invariant transport-kernel allocation rule Palm measure Abelian group mass-stationarity


Last, Günter; Thorisson, Hermann. Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37 (2009), no. 2, 790--813. doi:10.1214/08-AOP420.

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  • [1] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508 (electronic).
  • [2] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66.
  • [3] Chatterjee, S., Peled, R., Peres, Y. and Romik, R. (2008). Gravitational allocation to Poisson points. Ann. Math. To appear.
  • [4] Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 141–152.
  • [5] Geman, D. and Horowitz, J. (1975). Random shifts which preserve measure. Proc. Amer. Math. Soc. 49 143–150.
  • [6] Harris, T. E. (1971). Random measures and motions of point processes. Z. Wahrsch. Verw. Gebiete 18 85–115.
  • [7] Heveling, M. and Last, G. (2005). Characterization of Palm measures via bijective point-shifts. Ann. Probab. 33 1698–1715.
  • [8] Heveling, M. and Last, G. (2007). Point shift characterization of Palm measures on Abelian groups. Electron. J. Probab. 12 122–137 (electronic).
  • [9] Holroyd, A. E. and Liggett, T. M. (2001). How to find an extra head: Optimal random shifts of Bernoulli and Poisson random fields. Ann. Probab. 29 1405–1425.
  • [10] Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Comm. Probab. 8 17–27 (electronic).
  • [11] Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Probab. 33 31–52.
  • [12] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [13] Kallenberg, O. (2007). Invariant measures and disintegrations with applications to Palm and related kernels. Probab. Theory Related Fields 139 285–310.
  • [14] Last, G. (2006). Stationary partitions and Palm probabilities. Adv. in Appl. Probab. 38 602–620.
  • [15] Last, G. (2009). Modern random measures: Palm theory and related models. In New Perspectives in Stochastic Geometry (W. Kendall und I. Molchanov, eds.). Clarendon Press, Oxford. To appear.
  • [16] Last, G. (2009). Stationary random measures on homogeneous spaces. To appear.
  • [17] Last, G. and Thorisson, H. (2009). Characterization of mass-stationary by Bernoulli and Cox transports. To appear.
  • [18] Last, G. and Thorisson, H. (2008). Constructions of stationary and mass-stationary random measures. (In preparation.)
  • [19] Liggett, T. M. (2002). Tagged particle distributions or how to choose a head at random. In In and Out of Equilibrium (Mambucaba, 2000). Progr. Probab. 51 133–162. Birkhäuser, Boston.
  • [20] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester.
  • [21] Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete 9 36–58.
  • [22] Mecke, J. (1975). Invarianzeigenschaften allgemeiner Palmscher Maße. Math. Nachr. 65 335–344.
  • [23] Neveu, J. (1977). Processus ponctuels. In École D’Été de Probabilités de Saint-Flour, VI—1976. Lecture Notes in Mathematics 598 249–445. Springer, Berlin.
  • [24] Port, S. C. and Stone, C. J. (1973). Infinite particle systems. Trans. Amer. Math. Soc. 178 307–340.
  • [25] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. I: Theory. Springer, New York.
  • [26] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057–2064.
  • [27] Thorisson, H. (1999). Point-stationarity in d dimensions and Palm theory. Bernoulli 5 797–831.
  • [28] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
  • [29] Timár, Á. (2004). Tree and grid factors for general point processes. Electron. Comm. Probab. 9 53–59 (electronic).
  • [30] Zähle, U. (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. Probab. Theory Related Fields 80 79–100.