Invariance principles for renewal processes when only moments of low order exist

Abstract

Starting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419-1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived.Invariance principles strong approximations weak approximations renewal processes random sums Wiener process extreme value theorem