Namespaces
Variants
Actions

Difference between revisions of "Spectral function of a stationary stochastic process"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
''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''
+
{{TEX|done}}
 +
''spectral function of a homogeneous random field in an $n$-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.
+
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|stochastic process]] which is stationary in the wide sense, or of a [[Random field|random field]] in an $n$-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 $\lambda$, 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.

Latest revision as of 16:56, 30 July 2014

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.

How to Cite This Entry:
Spectral function of a stationary stochastic process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_function_of_a_stationary_stochastic_process&oldid=32589
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article