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Some local approximation properties of simple point processes
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  • Published: 05 December 2007

Some local approximation properties of simple point processes

  • Olav Kallenberg1 

Probability Theory and Related Fields volume 143, pages 73–96 (2009)Cite this article

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Abstract

For simple point processes ξ on a Borel space S, we prove some approximations involving conditional distributions, given that ξ hits a small set B. Beginning with general versions of some classical limit theorems, going back to the pioneering work of Palm and Khinchin, we proceed to prove that, under suitable regularity conditions, the contributions to B and B c are asymptotically conditionally independent. We further derive approximations in total variation of reduced Palm distributions and show that, when ξ hits some small sets B 1,..., B n , the corresponding restrictions are asymptotically independent. Next we give general versions of the asymptotic relations P{ξ B > 0} ~ Eξ B and prove some ratio limit theorems for conditional expectations E[η | ξ B > 0], valid even when Eξ is not σ-finite and the Palm distributions may fail to exist.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL, 36849, USA

    Olav Kallenberg

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  1. Olav Kallenberg
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Correspondence to Olav Kallenberg.

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Kallenberg, O. Some local approximation properties of simple point processes. Probab. Theory Relat. Fields 143, 73–96 (2009). https://doi.org/10.1007/s00440-007-0120-z

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  • Received: 04 December 2006

  • Revised: 12 October 2007

  • Published: 05 December 2007

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0120-z

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Keywords

  • Palm and Campbell measures
  • Hitting probabilities
  • Conditional independence
  • Ratio limit theorems
  • Differentiation of measures

Mathematics Subject Classification (2000)

  • 60G55
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