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

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

References

[1]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[2]	Yu.A. Rozanov, "Stationary random processes" , Holden-Day (1967) (Translated from Russian)