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

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.