Response to Comments on PCA Based Hurst Exponent Estimator for fBm Signals Under Disturbances

TL;DR

This paper addresses corrections to a PCA-based Hurst exponent estimator for fBm signals and explores the asymptotic behavior of eigenvalues in Gaussian processes with different series expansions.

Contribution

It provides a corrected proof for the PCA-based estimator and investigates the relationship between eigenvalue sequences of Gaussian processes with different Riesz bases.

Findings

Corrected the proof of the Hurst exponent estimator using orthogonal projection.

Analyzed the asymptotic behavior of eigenvalues in Gaussian processes with different series expansions.

Provided insights into the eigenvalue sequence relationships in Gaussian processes.

Abstract

In this response, we try to give a repair to our previous proof for PCA Based Hurst Exponent Estimator for fBm Signals by using orthogonal projection. Moreover, we answer the question raised recently: If a centered Gaussian process $G_t$ admits two series expansions on different Riesz bases, we may possibly study the asymptotic behavior of one eigenvalue sequence from the knowledge on the asymptotic behaviors of another.