  In this paper we study the weak convergence of self-normalized partial sum
processes in the Skorokhod M1 topology for sequences of random variables which
exhibit clustering of large values of the same sign. We show that for
stationary regularly varying sequences with such properties, their
corresponding properly centered self-normalized partial sums processes converge
to a stable Levy process. The convergence is established in the space of cadlag
functions endowed with Skorohod's M1 topology, which is more suitable
especially for cases in which the standard J1 topology fails to induce weak
convergence of joint stochastic functionals.
