Spectral function of a stationary stochastic process

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

A function of the frequency $\lambda$, or of a wave vector $\lambda=(\lambda_1,\ldots,\lambda_n)$, 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 $n$-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 $\lambda$, and the class of spectral functions of homogeneous random fields coincides with the class of functions in $n$ variables $\lambda_1,\ldots,\lambda_n$ differing only by a non-negative constant multiplier from $n$-dimensional probability distribution functions.

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Spectral function of a stationary stochastic process. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article