A Multivariate Functional Limit Theorem in Weak M1 Topology

TL;DR

This paper establishes a new functional limit theorem for weakly dependent regularly varying sequences of random vectors in the space of cadlag functions with weak M1 topology, extending classical Skorohod topology.

Contribution

It introduces a novel limit theorem in weak M1 topology for multivariate sequences, expanding the scope of functional limit theorems in dependent settings.

Findings

The convergence occurs in the space of R^d valued cadlag functions with weak M1 topology.

The extension of Skorohod's M1 topology is necessary for the theorem.

Illustrative examples demonstrate the applicability of the new 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 R^d valued c\`{a}dl\`{a}g functions endowed with the so-called weak M1 topology. The theory is illustrated on two examples. In particular, we demonstrate why such an extension of Skorohod's M1 topology is actually necessary for the limit theorem to hold.