Asymptotic expansions for the variance of stopping times in nonlinear renewal theory

Abstract

We treat the problem of finding asymptotic expansions for the variance of stopping times for Wiener processes with positive drift (continuous time case) as well as sums of i.i.d. random variables with positive mean (discrete time case). Carrying over the setting of nonlinear renewal theory to Wiener processes, we obtain an asymptotic expansion up to vanishing terms in the continuous time case. Applying the same methods to sums of i.i.d. random variables, we also provide an expansion in the discrete time case up to terms of order o( b 1/2) where the leading term is of order O( b), as b → ∞. The possibly unbounded term is the covariance of nonlinear excess and stopping time.