Asymptotic theory of mixed time averages and kth-order cyclic-moment and cumulant statistics

We generalize Parzen´s (1961) analysis of “asymptotically stationary” processes to mixtures of deterministic, stationary, nonstationary, and generally complex time series. Under certain mixing conditions expressed in terms of joint cumulant summability, we show that time averages of such mixtures converge in the mean-square sense to their ensemble averages. We additionally show that sample averages of arbitrary orders are jointly complex normal and provide their covariance expressions. These conclusions provide us with statistical tools that treat random and deterministic signals on a common framework and are helpful in defining generalized moments and cumulants of mixed processes. As an important consequence, we develop consistent and asymptotically normal estimators for time-varying, and cyclic-moments and cumulants of kth-order cyclostationary processes and provide computable variance expressions. Some examples are considered to illustrate the salient features of the analysis