Weak Convergence for Self-Normalized Partial Sum Processes in the Skorokhod M1 Topology with Applications to Regularly Varying Time Series

TL;DR

This paper establishes the weak convergence of self-normalized partial sum processes in the Skorokhod M1 topology for certain heavy-tailed, regularly varying time series with clustering of large values, demonstrating convergence to a stable Levy process.

Contribution

It introduces a new weak convergence result in the M1 topology for self-normalized sums of dependent, heavy-tailed sequences with clustering, extending previous methods.

Findings

Convergence to stable Levy processes for regularly varying sequences.

M1 topology is suitable where J1 topology fails.

Applicable to dependent sequences with clustering of large values.

Abstract

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.