Convergence rates for stopped random sums

Abstract

In this article we derive rates of convergence to normality for randomly stopped sums of suitably normalized i.d.d. random vectors in R k . The summation indices τ n are assumed to be stopping times — an assumption which is often fulfilled in interesting applications such as sequential analysis, random walk problems and actuarial mathematics — for which τ n / n converges in probability to a limit function τ satisfying the moment condition ∫(log(τ ∨ e)) ε d P < ∞ for some ε \\s#62;0. Examples show that the convergence rates presented are sharp and that the moment condition imposed on the limit function τ cannot be dispensed with.