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Self-Decomposable Laws from Continuous Branching Processes

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  • Anthony G. Pakes

    (University of Western Australia)

Abstract

The martingale limit law of the supercritical continuous time and state branching process either is compound Poisson or self-decomposable. This paper explores some general aspects of the latter case. A fundamental question for the latter case is whether the cumulant function of the martingale limit is a Thorin–Bernstein function. We make some progress by showing that it is special Bernstein if the cumulant function of the generating subordinator is special Bernstein. A specific parametric family of martingale limit cumulant functions is shown to be Thorin–Bernstein. Two complementary proofs of this fact are offered, one of which entirely avoids complex variable issues. The principal Lambert W-function is a boundary case of this family, thereby giving a new proof that it too is Thorin–Bernstein. Tail estimates of the distribution functions for this family are derived along with the right-hand tail and integral representations of their Lévy densities.

Suggested Citation

  • Anthony G. Pakes, 2020. "Self-Decomposable Laws from Continuous Branching Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 361-395, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-019-00886-0
    DOI: 10.1007/s10959-019-00886-0
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    References listed on IDEAS

    as
    1. Pakes, Anthony G., 2018. "The Lambert W function, Nuttall’s integral, and the Lambert law," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 53-60.
    2. Bingham, N. H., 1976. "Continuous branching processes and spectral positivity," Stochastic Processes and their Applications, Elsevier, vol. 4(3), pages 217-242, August.
    3. Pakes, Anthony G., 2013. "Limit laws for UGROW random graphs," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2607-2614.
    4. Lennart Bondesson, 2015. "A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1063-1081, September.
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    Cited by:

    1. Yang, Hairuo, 2023. "On the law of terminal value of additive martingales in a remarkable branching stable process," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 361-376.

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