Modeling temporally uncorrelated components of complex-valued stationary processes

Lietzén, Niko; Viitasaari, Lauri; Ilmonen, Pauliina

doi:10.15559/21-VMSTA190

Modern Stochastics: Theory and Applications

Modeling temporally uncorrelated components of complex-valued stationary processes

Volume 8, Issue 4 (2021), pp. 475–508

Niko Lietzén

Lauri Viitasaari Pauliina Ilmonen

https://doi.org/10.15559/21-VMSTA190

Pub. online: 10 November 2021 Type: Research Article

Open Access

Received
4 June 2021

Revised
6 October 2021

Accepted
6 October 2021

Published
10 November 2021

Abstract

A complex-valued linear mixture model is considered for discrete weakly stationary processes. Latent components of interest are recovered, which underwent a linear mixing. Asymptotic properties are studied of a classical unmixing estimator which is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag τ. The main contributions are asymptotic results that can be applied to a large class of processes. In related literature, the processes are typically assumed to have weak correlations. This class is extended, and the unmixing estimator is considered under stronger dependency structures. In particular, the asymptotic behavior of the unmixing estimator is estimated for both long- and short-range dependent complex-valued processes. Consequently, this theory covers unmixing estimators that converge slower than the usual $\sqrt{T}$ and unmixing estimators that produce non-Gaussian asymptotic distributions. The presented methodology is a powerful preprocessing tool and highly applicable in several fields of statistics.

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Keywords

Asymptotic theory blind source separation long-range dependency multivariate analysis noncentral limit theorems

MSC2010

62H12 60F05 60G15 60G10 94A12 94A08

Funding

N. Lietzén gratefully acknowledges financial support from the Emil Aaltonen Foundation (Grant 180144 N) and from the Academy of Finland (Grant 321968).

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