Weak convergence of the function-indexed integrated periodogram for infinite variance processes

TL;DR

This paper investigates the weak convergence of the integrated periodogram for linear processes with symmetric alpha-stable innovations, revealing stable process limits and differing methods for various alpha ranges.

Contribution

It provides the first comprehensive analysis of the weak limits of function-indexed integrated periodograms for infinite variance processes, including new methods for different alpha regimes.

Findings

Weak limits are alpha-stable processes with Fourier series representations.

Different techniques are used for alpha in (0,1) and [1,2), including entropy conditions.

Results apply to infinite mean quadratic forms with Toeplitz matrices.

Abstract

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $\alpha$-stable innovations. Under suitable summability conditions on the series of the Fourier coefficients of the index functions, we show that the weak limits constitute $\alpha$-stable processes which have representations as infinite Fourier series with i.i.d. $\alpha$-stable coefficients. The cases $\alpha\in(0,1)$ and $\alpha\in[1,2)$ are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case $\alpha\in(0,1)$, entropy conditions are needed for $\alpha\in[1,2)$ to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.