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2 results

  • Convergence Rates in the Implicit Renewal Theorem on Trees

  • Part of
  • Predrag R. Jelenković, Mariana Olvera-Cravioto
  • Journal:
    Journal of Applied Probability / Volume 50 / Issue 4 / December 2013
    Published online by Cambridge University Press:
    30 January 2018, pp. 1077-1088
    Print publication:
    December 2013
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  • We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

  • Implicit Renewal Theory and Power Tails on Trees

  • Part of
  • Predrag R. Jelenković, Mariana Olvera-Cravioto
  • Journal:
    Advances in Applied Probability / Volume 44 / Issue 2 / June 2012
    Published online by Cambridge University Press:
    04 January 2016, pp. 528-561
    Print publication:
    June 2012
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  • We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C1, C2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C1, C2,…); here ‘∨’ denotes the maximum operator.