Spectral function of a stationary stochastic process

spectral function of a homogeneous random field in an -dimensional space, spectral distribution function

A function of the frequency , or of a wave vector , respectively, occurring in the spectral decomposition of the covariance function of a stochastic process which is stationary in the wide sense, or of a random field in an -dimensional space which is homogeneous in the wide sense, respectively (cf. Spectral decomposition of a random function). The class of spectral functions of stationary stochastic processes coincides with that of all bounded monotonically non-decreasing functions , and the class of spectral functions of homogeneous random fields coincides with the class of functions in variables differing only by a non-negative constant multiplier from -dimensional probability distribution functions.