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Poisson convergence and poisson processes with applications to random graphs

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  • Janson, Svante

Abstract

We give a new sufficient condition for convergence to a Poisson distribution of a sequence of sums of dependent variables. The condition allows each summand to depend strongly on a few of the other variables and to depend weakly on the remaining ones. As a consequence we obtain sufficient conditions for the convergence of point processes, constructed as sets of (weakly) dependent random points in some space S, to a Poisson process. The main applications are to random graph theory. In particular, we solve the problem (proposed by Erdös) of finding the size of the first cycle in a random graph.

Suggested Citation

  • Janson, Svante, 1987. "Poisson convergence and poisson processes with applications to random graphs," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 1-30.
    • Handle: RePEc:eee:spapps:v:26:y:1987:i::p:1-30
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    Cited by:

    1. Oleg Seleznjev & Bernhard Thalheim, 2003. "Average Case Analysis in Database Problems," Methodology and Computing in Applied Probability, Springer, vol. 5(4), pages 395-418, December.
    2. Schulte, Matthias & Thäle, Christoph, 2012. "The scaling limit of Poisson-driven order statistics with applications in geometric probability," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 4096-4120.

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