Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation

TL;DR

This paper derives bounds for the covariance of functions of infinite variance stable variables, enabling new central limit theorems and wavelet-based estimation methods for stable processes.

Contribution

It introduces novel covariance bounds involving dependence measures for functions of stable variables, extending CLTs to unbounded functions and applications in wavelet estimation.

Findings

Established covariance bounds for functions of stable variables.

Proved CLTs for unbounded functions of stable moving averages.

Applied results to wavelet-based estimators in fractional stable motions.

Abstract

We establish bounds for the covariance of a large class of functions of infinite variance stable random variables, including unbounded functions such as the power function and the logarithm. These bounds involve measures of dependence between the stable variables, some of which are new. The bounds are also used to deduce the central limit theorem for unbounded functions of stable moving average time series. This result extends the earlier results of Tailen Hsing and the authors on central limit theorems for bounded functions of stable moving averages. It can be used to show asymptotic normality of wavelet-based estimators of the self-similarity parameter in fractional stable motions.