Correlations and spectra for non-stationary random functions

Introduction. The definition of the correlation function and the spectrum of a stationary random function is now classical, but in many applications one feels the need to extend this definition to random functions, which, although non-stationary, are in some sense nearly stationary. We suggest, therefore, for the definition of the correlation of a random function whose covariance T (t, s) is known, the limit (Li) *W-tᵗIP * (*-? * +) #> if this limit exists for every h. The spectrum S (\) can then be obtained from R (h) 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.