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Improved criteria for distributional convergence of point processes

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  • Kallenberg, Olav

Abstract

In earlier work by the author, the convergence in distribution of a sequence of point processes towards a simple limit has been characterized by three conditions, where the first ensures convergence of the supports, the second that the limit of every convergent subsequence is simple, and the third that the sequence is tight. The purpose of this note is to strengthen the mentioned result, by showing that the tightness follows from the other two conditions. Similar results are obtained for convergence of random measures towards a diffuse limit and of row-sums in null-arrays towards an infinitely divisible limit.

Suggested Citation

  • Kallenberg, Olav, 1996. "Improved criteria for distributional convergence of point processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 93-102, November.
    • Handle: RePEc:eee:spapps:v:64:y:1996:i:1:p:93-102
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    Cited by:

    1. Tyran-Kaminska, Marta, 2010. "Convergence to Lévy stable processes under some weak dependence conditions," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1629-1650, August.
    2. Asmussen, Soren & Ivanovs, Jevgenijs & Segers, Johan, 2017. "On the longest gap between power-rate arrivals," LIDAM Discussion Papers ISBA 2017014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Peterson, Lisa D., 2011. "Convergence in distribution of point processes on Polish spaces to a simple limit," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1859-1861.

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