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Weak Convergence Results for Multiple Generations of a Branching Process

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  • James Kuelbs

    (University of Wisconsin)

  • Anand N. Vidyashankar

    (Cornell University)

Abstract

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C 0[0,1])∞, with the product topology, or in Banach subspaces of (C 0[0,1])∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.

Suggested Citation

  • James Kuelbs & Anand N. Vidyashankar, 2011. "Weak Convergence Results for Multiple Generations of a Branching Process," Journal of Theoretical Probability, Springer, vol. 24(2), pages 376-396, June.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:2:d:10.1007_s10959-009-0266-y
    DOI: 10.1007/s10959-009-0266-y
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    References listed on IDEAS

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    1. Scott, D. J., 1978. "A central limit theorem for martingales and an application to branching processes," Stochastic Processes and their Applications, Elsevier, vol. 6(3), pages 241-252, February.
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