Fourier Representations of Spectral Densities in Long-Memory Processes

TL;DR

This paper investigates the Fourier series behavior of spectral densities in long-memory processes, constructing a specific example to highlight divergence issues and comparing empirical and theoretical autocovariances.

Contribution

It constructs a long-memory process with divergent Fourier series of spectral density, emphasizing the need for careful handling in spectral analysis of such processes.

Findings

Spectral density Fourier series can diverge almost everywhere in long-memory processes.

Under regular variation assumptions, Fourier series converge except possibly at zero.

Constructed process can be simulated and compared with theoretical autocovariances.

Abstract

In this article, we aim to further clarify certain subtle aspects of processes that exhibit long memory in the second-order sense. We construct a long-memory stochastic sequence, in the sense that the series of absolute autocovariances diverges, whose spectral density has an almost everywhere unboundedly divergent Fourier series. This suggests that the Fourier series of the spectral density of a generic long-range dependent process, one for which nothing is known except that its autocovariances are not absolutely summable, should be handled with great care. On the other hand, it is known that if one assumes regularly varying behavior for the autocovariances or the spectral density, along with suitable conditions on the associated slowly varying function, then the Fourier series of the spectral density converges everywhere, except possibly at 0. The process we construct can easily be simulated, and we compare its empirical and theoretical autocovariances.