Let (X0,Y0), (X1,Y1),...be a sequence of independent two-dimensional random vectors such that (X1,Y1), (X2, Y2),...are i.i.d. Let {(Sn,Un)}nÅ≥0 be the associated sum process, and define T(t)=inf{n≥0:Sn>t) for t≥0. Under suitable conditions on (X0,Y0) and (X1,Y1) the paper derives expansions up to vanishing terms, as t⇒•, for EUTÅ(tÅ), VarUTÅ(tÅ) and Cov(UTÅ(tÅ),T(t)). Corresponding results will be obtained for EUNÅ(tÅ), VarUNÅ(tÅ) and Cov(UNÅ(tÅ),N(t)) when X0,X1 are both almost surely non-negative and N(t)=sup{n≥0:Sn•t}.
