Convergence to L\'evy stable processes under some weak dependence conditions

TL;DR

This paper establishes necessary and sufficient conditions for the convergence of partial sum processes of stationary sequences to Le9vy stable processes, especially under strong mixing conditions, in the Skorohod space.

Contribution

It provides a comprehensive characterization of convergence criteria to Le9vy stable processes, including new conditions for strongly mixing sequences.

Findings

Identifies necessary and sufficient conditions for convergence.

Provides sufficient conditions under strong mixing assumptions.

Clarifies the role of dependence structures in convergence to stable processes.

Abstract

For a strictly stationary sequence of random vectors in $\mathbb{R}^d$ we study convergence of partial sum processes to L\'evy stable process in the Skorohod space with $J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.