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

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  • Alsmeyer, Gerold
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    Abstract

    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 55 (2001)
    Issue (Month): 2 (November)
    Pages: 163-171

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    Handle: RePEc:eee:stapro:v:55:y:2001:i:2:p:163-171

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    Related research

    Keywords: Random walks First passage times Nonlinear renewal theory Weak convergence;

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