# Difference between revisions of "Spectral density"

of a stationary stochastic process or of a homogeneous random field in $n$- dimensional space

The Fourier transform of the covariance function of a stochastic process which is stationary in the wide sense (cf. Stationary stochastic process; Random field, homogeneous). Stationary stochastic processes and homogeneous random fields for which the Fourier transform of the covariance function exists are called processes with a spectral density.

Let

$$X ( t) = \{ X _ {k} ( t) \} _ {k=} 1 ^ {n}$$

be an $n$- dimensional stationary stochastic process, and let

$$X ( t) = \int\limits e ^ {i t \lambda } \Phi ( d \lambda ) ,\ \Phi = \ \{ \Phi _ {k} \} _ {k=} 1 ^ {n}$$

be its spectral representation ( $\Phi _ {k}$ is the spectral measure corresponding to the $k$- th component $X _ {k} ( t)$ of the multi-dimensional stochastic process $X ( t)$). The range of integration is $- \pi \leq \lambda \leq \pi$ in the case of discrete time $t$, and $- \infty < \lambda < + \infty$ in the case of continuous time $t$. The process $X ( t)$ has a spectral density

$$f ( \lambda ) = \{ f _ {k,l} ( \lambda ) \} _ {k,l=} 1 ^ {n} ,$$

if all the elements

$$F _ {k,l} ( \Delta ) = {\mathsf E} \Phi _ {k} ( \Delta ) \overline{ {\Phi _ {l} ( \Delta ) }}\; ,\ \ k , l = {1 \dots n } ,$$

of the spectral measure $F = \{ F _ {k,l} \} _ {k,l=} 1 ^ {n}$ are absolutely continuous and if

$$f _ {k,l} ( \lambda ) = \ \frac{F _ {k,l} ( d \lambda ) }{d \lambda } .$$

In particular, if the relation

$$\sum _ {l = - \infty } ^ \infty | B _ {k,l} ( t) | < \infty ,\ \ k , l = {1 \dots n } ,$$

holds for $X ( t)$, $t = 0 , \pm 1 \dots$ where

$$B ( t) = \{ B _ {k,l} ( t) \} _ {k,l=} 1 ^ {n} = \ \{ {\mathsf E} X _ {k} ( t + s ) \overline{ {X _ {l} ( s) }}\; \} _ {k,l=} 1 ^ {n}$$

is the covariance function of $X ( t)$, then $X ( t)$ has a spectral density and

$$f _ {k,l} ( \lambda ) = ( 2 \pi ) ^ {-} 1 \sum _ {t = - \infty } ^ \infty B _ {k,l} ( t) \mathop{\rm exp} \{ - i \lambda t \} ,$$

$$- \infty < \lambda < \infty ,\ k , l = {1 \dots n } .$$

The situation is similar in the case of processes $X ( t)$ in continuous time $t$. The spectral density $f ( \lambda )$ is sometimes called the second-order spectral density, in contrast to higher spectral densities (see Spectral semi-invariant).

A homogeneous $n$- dimensional random field $X ( t _ {1} \dots t _ {n} )$ has a spectral density $f ( \lambda _ {1} \dots \lambda _ {n} )$ if its spectral resolution $F ( \lambda _ {1} \dots \lambda _ {n} )$ possesses the property that its mixed derivative $\partial ^ {n} F / \partial \lambda _ {1} \dots \partial \lambda _ {n}$ exists almost-everywhere, and then

$$f ( \lambda _ {1} \dots \lambda _ {n} ) = \ \frac{\partial ^ {n} F }{\partial \lambda _ {1} \dots \partial \lambda _ {n} }$$

and

$$F ( \lambda _ {1} \dots \lambda _ {n} ) = \ \int\limits _ { \lambda _ {01} } ^ { {\lambda _ 1 } } \dots \int\limits _ { \lambda _ {0n} } ^ { {\lambda _ n } } f ( \mu _ {1} \dots \mu _ {n} ) \ d \mu _ {1} \dots d \mu _ {n} + \textrm{ const } .$$

How to Cite This Entry:
Spectral density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_density&oldid=48758
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article