Spectral analysis of the covariance of the almost periodically correlated processes

This paper deals with the spectrum of the almost periodically correlated (APC) processes defined on . It is established that the covariance kernel of such a process admits a Fourier series decomposition, K(s+t, s) , whose coefficient functions b[alpha] are the Fourier transforms of complex measures m[alpha], [alpha][set membership, variant], which are absolutely continuous with respect to the measure mo. Considering the APC strongly harmonizable processes, the spectral covariance of the process can be expressed in terms of these complex measures m[alpha]. The usual estimators for the second order situation can be modified to provide consistent estimators of the coefficient functions b[alpha] from a sample of the process. Whenever the measures m[alpha] are absolutely continuous with respect to the Lebesgue measure, so m[alpha](d[lambda])=f[alpha]([lambda]) d[lambda], the estimation of the corresponding density functions f[alpha] is considered. Under hypotheses on the covariance kernel K and on the coefficient functions b[alpha], we establish rates of convergence in quadratic mean and almost everywhere of these estimators.