A functional limit theorem for self-normalized partial sum processes in the $M_{1}$ topology

TL;DR

This paper establishes a self-normalized functional limit theorem for stationary sequences with heavy tails, demonstrating convergence in the Skorokhod M1 topology under certain dependence conditions.

Contribution

It introduces a new limit theorem for self-normalized partial sums in the M1 topology for stationary sequences with regular variation.

Findings

Proves convergence of self-normalized partial sums in the M1 topology.

Applicable to sequences with heavy tails and weak dependence.

Extends existing limit theorems to a broader class of processes.

Abstract

For a stationary sequence of random variables we derive a self-normalized functional limit theorem under joint regular variation with index $\alpha \in (0,2)$ and weak dependence conditions. The convergence takes place in the space of real-valued cadlag functions on $[0,1]$ with the Skorokhod $M_{1}$ topology.