The higher order theory of generalized almost-cyclostationary time series

In this paper, the class of generalized almost-cyclostationary (GACS) time series is introduced. Time series belonging to this class are characterized by multivariate statistical functions that are almost-periodic functions of time whose Fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. Almost-cyclostationary (ACS) time series turn out to be the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the above-mentioned sets is countable. The higher order characterization of GACS time series in the strict and wide sense is provided. It is shown that the characterization in terms of cyclic moment and cumulant functions is inadequate for those GACS time series that are not ACS. Then, generalized cyclic moment and cumulant functions (in both the time and frequency domains) are introduced. Finally, the problem of estimating the introduced generalized cyclic statistics is addressed, and two examples of GACS time series are considered.