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-t^tIP *(*-?* + $)#> 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.