Asymptotic normality of wavelet estimators of the memory parameter for linear processes

TL;DR

This paper proves that wavelet-based estimators for the memory parameter in linear processes are asymptotically normal, providing explicit variance formulas, applicable to processes with various memory types.

Contribution

It establishes the asymptotic normality of wavelet estimators for the memory parameter in linear processes, including non-Gaussian cases, with explicit variance expressions.

Findings

Wavelet estimators are asymptotically normal as sample size increases.

Explicit limit variance formulas are derived for the estimators.

Results apply to processes with long, short, or negative memory.

Abstract

We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size $n\to\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$.