On the optimality of strong approximation rates for compound renewal processes

The optimality of certain approximation rates appearing in strong invariance principles for partial sums indexed by a renewal process is discussed. The results extend and unify earlier work on the best rates in the invariance principles for renewal counting processes. The motivation for this note came from a recent approximation of compound renewal processes due to Csörgo, Deheuvels and Horváth (1987), which is presented here in a slightlty extended version.compound renewal processes Wiener process strong invariance principles optimal approximation rates