On the Fourier expansion of stationary random processes
Proceedings of the American Mathematical Society
The purpose of this note is to demonstrate that in the Fourier expansion of a quasistationary random continuous process with continuous covariance function, the amplitudes of the frequency components do not possess the desirable property of being mutually uncorrelated unless the process degenerates to a single random variable in its range of definition. Specifically we consider a real-valued continuous random process x(t) observed during a time interval T. We assume that the mean and covariance
... mean and covariance function of x(t) exist, so with no loss in generality we assume that Ex(t) = 0 and Ex(s)xit) = r(s -t), where r(r) is an even function of t and is continuous at r=0 with r(0) = l. It is then well known [l] that r(r) is continuous for all r. Consider the Fourier expansion of x(¿) given by in which l.i.m. denotes limit in the mean constructed from the covariance function r(s-t). In the engineering literature  on random noise it is assumed quite frequently in the Fourier expansion of a quasistationary process that £(a"am)=0 for nj^m, and E(anbm)=0 for all positive integers n and m. What we show is that the only quasistationary process for which this can hold even for one pair of integers n, m (nj¿m) and for all finite observation times1 T is the trivial process in which r(s-t) = 1 for all s and t. It is well known that there exist quasistationary processes with discontinuous covariance functions for which the property holds that the amplitudes are mutually uncorrelated. The simplest example of one of these is a "pure white noise" which is characterized by Ex(t)2 = 1; £x(5)x(/)=0, for s^t. Moreover when the assumption of quasistationarity is dropped, and one considers processes for which the covariance function r(s, t) may not be of the form r(s-t), then there Received by the editors December 10, 1952. 1 It is easy to prove for sufficiently long observation times that as T increases, the correlation coefficient between any two different amplitudes in the Fourier expansion decreases and approaches zero as T approaches infinity. 564 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FOURIER EXPANSION OF STATIONARY RANDOM PROCESSES 565 exist random processes with continuous covariance function for which the amplitudes of the Fourier components are mutually uncorrelated. The best known example is the Wiener process, for which r(s, t) =dF(«) J -» in which F(w) is a bounded, nondecreasing function and is termed the spectral function of the process. We state and prove the following theorem : Theorem. Given a real valued quasistationary random continuous process with spectral function F(u) and variance f2KdF(w) = 1, and for which the Fourier expansion in equation (2) converges in the mean. The necessary and sufficient condition that for any pair of unequal positive integers n and m and every finite time interval T the amplitudes an, am, bn, bm satisfy the relations Eanam -Eanbm = Eambn = Ebnbm = Q is that F(co) = 0 for o) <0 and F(u) = 1 for w = 0. Proof. Since it is more convenient to work with the expansion given in (1) , we require the following lemma: Lemma. For any pair of unequal positive integers n and m, the relations Eanam = Eanbm = Eambn = Ebnbm = 0 are equivalent to the relations Ecncm = Ecnc-m = Q. The proof of the lemma is very simple and is obtained directly from the relations EcnCm = 4E[(anam + bnbm) + i(a"bm -ambn)], EcnC-m = 4£[(a"am -bnbm) + i( -anbm -amb")].