Functional Convergence of Linear Sequences in a non-Skorokhod Topology

TL;DR

This paper establishes a new functional limit theorem for linear sequences of random variables in a non-Skorokhod topology, extending previous results to more general coefficient conditions and demonstrating convergence to a linear fractional stable motion.

Contribution

It introduces a novel weak convergence result in the $S$-topology for linear sequences with sign-changing coefficients, extending prior work to broader coefficient classes.

Findings

Convergence of partial sums to a linear fractional stable motion in the $S$-topology.

Extension of Avram and Taqqu's results to sign-changing coefficients.

Validation through examples and computer simulations.

Abstract

In this article, we prove a new functional limit theorem for the partial sum sequence $S_{[nt]}=\sum_{i=1}^{[nt]}X_i$ corresponding to a linear sequence of the form $X_i=\sum_{j \in \bZ}c_j \xi_{i-j}$ with i.i.d. innovations $(\xi_i)_{i \in \bZ}$ and real-valued coefficients $(c_j)_{j \in \bZ}$. This weak convergence result is obtained in space $\bD[0,1]$ endowed with the $S$-topology introduced in Jakubowski (1992), and the limit process is a linear fractional stable motion (LFSM). One of our result provides an extension of the results of Avram and Taqqu (1992) to the case when the coefficients $(c_j)_{j \in \bZ}$ may not have the same sign. The proof of our result relies on the recent criteria for convergence in Skorokhod's $M_1$-topology (due to Louhichi and Rio (2011)), and a result which connects the weak $S$-convergence of the sum of two processes with the weak $M_1$-convergence of the two individual processes. Finally, we illustrate our results using some examples and computer simulations.