Mean-square consistency of the $f$-truncated $\text{M}^2$-periodogram

TL;DR

This paper introduces an $f$-truncated periodogram for estimating the multivariate spectral density of stationary processes, demonstrating its asymptotic consistency and practical effectiveness through simulations.

Contribution

It proposes a new $f$-truncated periodogram estimator and proves its mean-square consistency for multivariate spectral density estimation.

Findings

Estimator is asymptotically consistent.

Simulation results confirm effectiveness.

Applicable to multiple problems in spectral analysis.

Abstract

The paper deals with the problem of estimating the M$^2$ (i.e. multivariate and multidimensional) spectral density function of a stationary random process or random field. We propose the $f$-truncated periodogram, i.e. a truncated periodogram where the truncation point is a suitable function $f$ of the sample size. We discuss the asymptotic consistency of the estimator and we provide three concrete problems that can be solved using the proposed approach. Simulation results show the effectiveness of the procedure.