Almost Periodic Variances

Abstract

Second-order stationary random sequences, especially if Gaussian, may be of statistical interest because their covariances may, under certain conditions, be consistently estimated with finite realizations of the sequences. It is shown that there is a class of non-stationary random sequences, namely sequences of orthogonal random variables with zero means and variances $f_t$ which form a uniformly almost periodic sequence, which are of statistical interest, at least in the Gaussian case, in the following sense. $f_t$ admits a generalized Fourier series expansion, and the Fourier coefficients $\gamma_s$ of this expansion can be consistently estimated with finite realizations of the sequences. In certain situations, the nonconstant variance sequence $f_t$ may be directly estimated. The sequences of interest may be converted, via Fourier transformations, into second-order stationary random functions, and the Fourier coefficients $\gamma_s$, of the expansion of $f_t$, are shown to form a sequence of stationary covariances. A multiplicative representation is given for the nonstationary sequences considered.