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

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.