Functional limit theorems for cumulative processes and stopping times

Summary Consider a cumulative regenerative process with increments between regeneration points being i.i.d. r.v.'s. Let the d.f. of those increments belong to the domain of attraction of a stable distribution with exponent less than two. A functional limit theorem in the Skorohod M ₁-topology is proved for this process. The M ₁-topology is more useful than the J ₁-topology in this case, because it allows the cumulative process to be continuous.The second part of the paper concerns a stopping time process, (t)--inf(s>0:w(s)>tg(s)), where w(t) is a process with positive drift for which a functional limit theorem holds and g(t)=t ^p L(t) with 0p<1 and L(t) varying slowly at infinity. Weak convergence for the process (t) is proved under certain conditions in the J ₁- and M ₁-topologies.