Correlations and spectra for non-stationary random functions

Abstract

in the classical way. We are led to the above definition of the correlation function R(h) by the following considerations: we determine the sample-correlation from a truncated sample of the random function; we then obtain a sub-correlation, RT(h), of the random function (defined as the correlation of the truncated random function) by averaging the sample correlations; finally, the correlation R(h) is defined by (1.1) as the limit of RT(h), if this limit exists. The function R(h), so defined, has all the properties of a correlation function. If the random function is stationary (wide sense) [4, p. 95-96], our definition coincides with the classical definition. The estimation of the correlation of a stationary random function has been considered extensively in the literature, particularly by U. Grenander and M. Rosenblatt [6], R. B. Blackman and J. W. Tukey [1], and E. Parzen [13, 14]. In order to evaluate how good the estimate R(h) is from the samplecorrelations pT(h, w), which are the only experimental observables, we compute the variance of the random variables pT(h, w) about RT(h), and then we compute (for a fixed h) an upper bound of R(h) - RT(h) for large T. This paper is especially concerned with the case in which the random function has a periodic covariance P (t + r, s + r) = r(t, s). To appreciate the scope of the above condition, let us note that it is always satisfied when the random function is a sum of two uncorrelated random functions, one being a stationary (wide sense) random function and the other a periodic random function. The last part of the paper is devoted to the estimate of R(h) for a non-stationary random step-function V(t, co), similar to the one introduced by N. Wiener,