Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On symmetrically distributed random measures


Author: Olav Kallenberg
Journal: Trans. Amer. Math. Soc. 202 (1975), 105-121
MSC: Primary 60G55
MathSciNet review: 0370751
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A random measure $ \xi $ defined on some measurable space $ (S,\mathcal{S})$ is said to be symmetrically distributed with respect to some fixed measure $ \omega $ on $ S$, if the distribution of $ (\xi {A_1}, \cdots ,\xi {A_k})$ for $ k \in N$ and disjoint $ {A_1}, \cdots ,{A_k} \in \mathcal{S}$ only depends on $ (\omega {A_1}, \cdots ,\omega {A_k})$. The first purpose of the present paper is to extend to such random measures (and then even improve) the results on convergence in distribution and almost surely, previously given for random processes on the line with interchangeable increments, and further to give a new proof of the basic canonical representation. The second purpose is to extend a well-known theorem of Slivnyak by proving that the symmetrically distributed random measures may be characterized by a simple invariance property of the corresponding Palm distributions.


References [Enhancements On Off] (What's this?)

  • [1] A. Benczur, On sequences of equivalent events and the compound Poisson process, Studia Sci. Math. Hungar. 3 (1968), 451-458. MR 39 #4905. MR 0243584 (39:4905)
  • [2] P. Billingsley, Convergence of probability measures, Wiley, New York, 1968. MR 38 #1718. MR 0233396 (38:1718)
  • [3] J. Hagberg, Approximation of the summation process obtained by sampling from a finite population, Teor. Verojatnost. i Primenen. 18 (1973), 790-803. MR 0328985 (48:7327)
  • [4] P. Jagers, On Palm probabilities, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 17-32. MR 0339330 (49:4089)
  • [5] O. Kallenberg, Characterization and convergence of random measures and point processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 (1973), 9-21. MR 0431374 (55:4374)
  • [6] -, A canonical representation of symmetrically distributed random measures, Mathematics and Statistics, Essays in Honour of Harald Bergström, Teknologtryck, Göteborg, 1973, pp. 41-48. MR 0372999 (51:9201)
  • [7] -, Canonical representations and convergence criteria for processes with interchangeable increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 (1973), 23-36. MR 0394842 (52:15641)
  • [8] -, Path properties of processes with independent and interchangeable increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28 (1974), 257-271. MR 0402901 (53:6715)
  • [9] G. Kummer and K. Matthes, Verallgemeinerung eines Satzes von Sliwnjak. II, III, Rev. Roumaine Math. Pures Appl. 15 (1970), 845-870, 1631-1642. MR 42 #5304; 44 #6024. MR 0270416 (42:5304)
  • [10] F. Papangelou, The Ambrose-Kakutani theorem and the Poisson process, Contributions to Ergodic Theory and Probability, Springer-Verlag, Berlin and New York, 1970, pp. 234-240.
  • [11] S. C. Port and C. J. Stone, Infinite particle systems, Trans. Amer. Math. Soc. 178 (1973), 307-340. MR 0326868 (48:5210)
  • [12] B. Rosén, Limit theorems for sampling from a finite population, Ark. Mat. 5 (1964), 383-424. MR 31 #1700.
  • [13] I. M. Slivnjak, Some properties of stationary flows of homogeneous random events, Theor. Probability Appl. 7 (1962), 336-341; ibid. 9 (1964), 168. MR 27 #832. MR 0150846 (27:832)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60G55

Retrieve articles in all journals with MSC: 60G55


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0370751-6
Keywords: Random measures and point processes, symmetric distributions, interchangeable increments, weak and strong convergence, invariance of Palm distributions
Article copyright: © Copyright 1975 American Mathematical Society