Asymptotic distribution of a robust wavelet-based NKK periodogram

TL;DR

This paper derives the asymptotic distribution of a robust wavelet-based NKK periodogram for long-memory time series with heavy tails, providing a theoretical foundation for spectral analysis using wavelets and LAD regression.

Contribution

It establishes the limiting distribution of the wavelet-based NKK periodogram under long-range dependence and heavy-tailed innovations, a novel theoretical result.

Findings

Convergence to a quadratic form in Gaussian vectors

Dependence on memory properties and wavelet filters

Supports robust spectral analysis of long-memory processes

Abstract

This paper investigates the asymptotic distribution of a wavelet-based NKK periodogram constructed from least absolute deviations (LAD) harmonic regression at a fixed resolution level. Using a wavelet representation of the underlying time series, we analyze the probabilistic structure of the resulting periodogram under long-range dependence. It is shown that, under suitable regularity conditions, the NKK periodogram converges in distribution to a nonstandard limit characterized as a quadratic form in a Gaussian random vector, whose covariance structure depends on the memory properties of the process and on the chosen wavelet filters. This result establishes a rigorous theoretical foundation for the use of robust wavelet-based periodograms in the spectral analysis of long-memory time series with heavy-tailed inovations.