Second-order approximations for certain stopped sums in extended renewal theory

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)}n.0 be the associated sum process, and define for t . 0 https://static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aartic le%3AS0001867800017031/resource/name/S0001867800017031_eqnU1.gif?pub-sta tus=live . Under suitable conditions on (X0, Y0) and (X1, Y1) we derive expansions up to vanishing terms, as t.., for EUT(t), Var UT(t) and Cov (UT(t), T(t)). Corresponding results will be obtained for EUN(t), Var UN(t) and Cov (UN(t), N(t)) when X0, .1 are both almost surely non-negative and https://static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aartic le%3AS0001867800017031/resource/name/S0001867800017031_eqnU2.gif?pub-sta tus=live.