Functional Convergence of Linear Sequences in a non-Skorokhod Topology

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.