Functional limit theory for the spectral covariance estimator

Abstract

Processes that exhibit repeatability in their kth-order moments are frequently studied in signal analysis. Such repeatability can be conveniently expressed with the help of almost periodicity. In particular, almost periodically correlated (APC) processes play an important role in the analysis of repeatable signals. This paper presents a study of asymptotic distributions of the estimator of the spectral covariance function for APC processes. It is demonstrated that, for a large class of APC processes, the functional central limit theorem holds.

Estimation for almost periodic processes

“…Lund et al [17] use an averaged squared coherence statistic to test existence of periodic correlation against stationarity. The case of a continuous parameter process {X(·)} is considered in [4] and [12].…”

Cyclostationarity: New trends and applications

“…are asymptotically (as T-1 and Δf -0 with TΔf -1) zeromean jointly complex normal [62,71,73,113,157,159] with asymptotic covariance and conjugate covariance matrices that can be obtained by specializing to the ACS case the results for SC processes in [254,Theorems 4.7.7,5.8.3]. An alternative estimator of the (conjugate) cyclic spectrum is the time-smoothed (conjugate) cyclic periodogram lim…”

Discrete Periodic Sampling with Jitter and Almost Periodically Correlated Processes

“…This condition is well known in the theory of almost periodically correlated processes (Gardner, 1994). For instance, the rate of convergence of the estimator of the spectral covariance or of the spectral density function, and the asymptotic normality of these estimators are obtained under this condition (Dehay, 1994;Dehay and Leśkow, 1995;Hurd and Leśkow, 1992). This condition is an identifiability condition.…”

“…From the properties of a * n (λ, τ ), which are straightforward consequences of those of the continuous time estimator a T (λ, τ ) (Dehay and Leśkow, 1995;Dehay and Monsan, 2003;Hurd and Leśkow, 1992), we are going to deduce the properties of a n (λ, τ ).…”