ASYMPTOTIC THEORY OF MIXED TIME AVERAGES AND KTH-ORDER CYCLIC-MOMENT AND CUMULANT STATISTICS

We generalize Parzen's analysis of ''asymptotically stationary'' proce sses to mixtures of deterministic, stationary, nonstationary, and gene rally complex time series. Under certain mixing conditions expressed i n terms of joint cumulant summability, we show that time averages of s uch mixtures converge in the mean-square sense to their ensemble avera ges. We additionally show that sample averages of arbitrary orders are jointly complex normal and provide their covariance expressions. Thes e conclusions provide us with statistical tools that treat random and deterministic signals on a common framework and are helpful in definin g generalized moments and cumulants of mixed processes. As an importan t consequence, we develop consistent and asymptotically normal estimat ors for time-varying, and cyclic- moments and cumulants of Kth-order c yclostationary processes and provide computable variance expressions. Some examples are considered to illustrate the salient features of the analysis.