View source for Spectral function of a stationary stochastic process

''spectral function of a homogeneous random field in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864201.png" />-dimensional space, spectral distribution function''

A function of the frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864202.png" />, or of a wave vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864203.png" />, respectively, occurring in the spectral decomposition of the covariance function of a [[Stochastic process|stochastic process]] which is stationary in the wide sense, or of a [[Random field|random field]] in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864204.png" />-dimensional space which is homogeneous in the wide sense, respectively (cf. [[Spectral decomposition of a random function|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864205.png" />, and the class of spectral functions of homogeneous random fields coincides with the class of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864206.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864207.png" /> differing only by a non-negative constant multiplier from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086420/s0864208.png" />-dimensional probability distribution functions.