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Thin tails of fixed points of the nonhomogeneous smoothing transform

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  • Alsmeyer, Gerold
  • Dyszewski, Piotr

Abstract

For a given random sequence (C,T1,T2,…), the smoothing transform S maps the law of a real random variable X to the law of ∑k≥1TkXk+C, where X1,X2,… are independent copies of X and also independent of (C,T1,T2,…). This law is a fixed point of S if X=d∑k≥1TkXk+C holds true, where =d denotes equality in law. Under suitable conditions including EC=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T1,T2,…) such that the canonical solution exhibits right and/or left Poissonian tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.

Suggested Citation

  • Alsmeyer, Gerold & Dyszewski, Piotr, 2017. "Thin tails of fixed points of the nonhomogeneous smoothing transform," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 3014-3041.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:9:p:3014-3041
    DOI: 10.1016/j.spa.2017.01.008
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    References listed on IDEAS

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    1. Alsmeyer, Gerold & Rösler, Uwe, 2003. "The best constant in the Topchii-Vatutin inequality for martingales," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 199-206, November.
    2. Baltrunas, A. & Daley, D. J. & Klüppelberg, C., 2004. "Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 237-258, June.
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