A Multivariate Functional Limit Theorem in Weak $$M_{1}$$ M 1 Topology

Abstract

We show a new functional limit theorem for weakly dependent regularly varying sequences of random vectors. As it turns out, the convergence takes place in the space of $$\mathbb R ^{d}$$ valued cadlag functions endowed with the so-called weak $$M_{1}$$ topology. The theory is illustrated on two examples. In particular, we demonstrate why such an extension of Skorohod’s $$M_1$$ topology is actually necessary for the limit theorem to hold.