A Proof on Asymptotics of Wavelet Variance of a Long Memory Process by Using Taylor Expansion

TL;DR

This paper proves that the log of wavelet variance for long memory processes grows linearly with scale, using Taylor expansion, highlighting their self-similar nature in low frequencies.

Contribution

It introduces a novel proof demonstrating the asymptotic linearity of the log wavelet variance for long memory processes via Taylor expansion.

Findings

Log wavelet variance is asymptotically proportional to scale.

The proof uses Taylor expansion to establish scale-invariance.

Results enhance understanding of long memory process characteristics.

Abstract

A long memory process has self-similarity or scale-invariant properties in low frequencies. We prove that the log of the scale-dependent wavelet variance for a long memory process is asymptotically proportional to scales by using the Taylor expansion of wavelet variances.