Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes

TL;DR

This paper establishes central limit theorems for wavelet packet coefficients of stationary random processes, showing their convergence to Gaussian processes with variances linked to the spectral density.

Contribution

It provides the first rigorous asymptotic analysis of wavelet packet coefficients for stationary processes, connecting spectral properties to Gaussian limits.

Findings

Wavelet packet coefficients converge to white Gaussian processes asymptotically.

Variance of the limit process relates to the spectral density at specific frequencies.

Results hold for any path in the wavelet packet decomposition tree.

Abstract

This paper provides central limit theorems for the wavelet packet decomposition of stationary band-limited random processes. The asymptotic analysis is performed for the sequences of the wavelet packet coefficients returned at the nodes of any given path of the $M$-band wavelet packet decomposition tree. It is shown that if the input process is centred and strictly stationary, these sequences converge in distribution to white Gaussian processes when the resolution level increases, provided that the decomposition filters satisfy a suitable property of regularity. For any given path, the variance of the limit white Gaussian process directly relates to the value of the input process power spectral density at a specific frequency.