Multichannel Deconvolution with Long Range Dependence: Upper bounds on the $L^p$-risk $(1 \le p < \infty)$

TL;DR

This paper studies multichannel deconvolution with long-memory errors, deriving upper bounds on convergence rates of wavelet estimators across different convolution scenarios and demonstrating the complexity introduced by fractional Brownian motion errors.

Contribution

It provides new upper bounds on the $L^p$-risk for wavelet estimators in multichannel deconvolution with long-range dependence errors, extending previous results to fractional Brownian motion cases.

Findings

Upper bounds on convergence rates for wavelet estimators across scenarios.

Long-range dependence significantly affects deconvolution performance.

Numerical results support theoretical bounds and highlight complexities.

Abstract

We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the $L^p$-risk $(1 \le p < \infty)$. Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.