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Two comparison theorems for nonlinear first passage times and their linear counterparts


  • Alsmeyer, Gerold


Let (Sn)n[greater-or-equal, slanted]0 be a zero-delayed nonarithmetic random walk with positive drift [mu] and ([xi]n)n[greater-or-equal, slanted]0 be a slowly varying perturbation process (see conditions (C.1)-(C.3) in Section 1). The results of this note are two weak convergence theorems for the difference [tau](t)-[nu](t), as t-->[infinity], where [tau](t)=inf{n[greater-or-equal, slanted]1: Sn>t} and [nu](t)=inf{n[greater-or-equal, slanted]1: Sn+[xi]n>t} denotes its nonlinear counterpart. The main result (Theorem 1) states the existence of a limit distribution for [tau](t)-[nu](t) providing the weak convergence of the [xi]n to a distribution [Lambda]. Two applications in sequential statistics are also given.

Suggested Citation

  • Alsmeyer, Gerold, 2001. "Two comparison theorems for nonlinear first passage times and their linear counterparts," Statistics & Probability Letters, Elsevier, vol. 55(2), pages 163-171, November.
  • Handle: RePEc:eee:stapro:v:55:y:2001:i:2:p:163-171

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    References listed on IDEAS

    1. Burman, Prabir, 1987. "Central limit theorem for quadratic forms for sparse tables," Journal of Multivariate Analysis, Elsevier, vol. 22(2), pages 258-277, August.
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