Gaussian process

A real stochastic process , , all finite-dimensional distributions of which are Gaussian, i.e. for any the characteristic function of the joint probability distribution of the random variables has the form

where is the mathematical expectation and

is the covariance function. The probability distribution of a Gaussian process is completely determined by its mathematical expectation and by the covariance function , . For any function and any positive-definite function there exists a Gaussian process with expectation and covariance function . A multi-dimensional stochastic process with vector values

is called Gaussian if the joint probability distributions of arbitrary variables

are Gaussian.

A complex Gaussian process , , is a process of the form

in which , jointly form a two-dimensional real Gaussian process. Regarding a complex Gaussian process one additional stipulation is imposed:

where

This condition is introduced in order to ensure the preservation of the equivalence between non-correlation and independence, which is a property of ordinary Gaussian random variables. It may be rewritten as follows:

where

is the covariance function of the process and

A linear generalized stochastic process , , on a linear space is called a generalized Gaussian process if its characteristic functional has the form

where is the mathematical expectation of the generalized process and

is its covariance functional.

Let be a Hilbert space with scalar product , . A random variable with values in is called Gaussian if , , is a generalized Gaussian process. The mathematical expectation is a continuous linear functional, while the covariance function is a continuous bilinear functional on the Hilbert space , and

where the positive operator is a nuclear operator, called the covariance operator. For any such and there exists a Gaussian variable such that the generalized process , , has expectation and covariance function .

Example. Let be a Gaussian process on the segment , let the process be measurable, and let also

Then almost-all the trajectories of , , will belong to the space of square-integrable functions on with the scalar product

The formula

defines a generalized Gaussian process on this space . The expectation and the covariance functional of the generalized process are expressed by the formulas

where and are, respectively, the expectation and the covariance function of the initial process on .

Almost-all the fundamental properties of a Gaussian process (the parameter runs through an arbitrary set ) may be expressed in geometrical terms if the process is considered as a curve in the Hilbert space of all random variables , , with the scalar product for which

and

Yu.A. Rozanov

Gaussian processes that are stationary in the narrow sense may be realized by way of certain dynamical systems (a shift in the space of trajectories [1]). The dynamical systems obtained (which are sometimes denoted as normal, on account of the resemblance to the normal probability distributions) are of interest as examples of dynamical systems with a continuous spectrum the properties of which can be more exhaustively studied owing to the decomposition of introduced in [4], [5]. The first actual examples of dynamical systems with "non-classical" spectral properties have been constructed in this way.

References

[1]	J.L. Doob, "Stochastic processes" , Chapman & Hall (1953)
[2]	I.A. Ibragimov, Yu.A. Rozanov, "Gaussian random processes" , Springer (1978) (Translated from Russian)
[3]	H. Cramér, M.R. Leadbetter, "Stationary and related stochastic processes" , Wiley (1967) pp. Chapts. 33–34
[4]	K. Itô, "Multiple Wiener integral" J. Math. Soc. Japan , 3 : 1 (1951) pp. 157–169
[5]	K. Itô, "Complex multiple Wiener integral" Japan J. Math. , 22 (1952) pp. 63–86

D.V. Anosov

Comments

A Gaussian process is sometimes called a normal process. See Stationary stochastic process for details about stationary Gaussian processes.

During the last twenty years hard work has been done by the American and French school in studying the regularity of the paths of a (real-valued) Gaussian process with respect to the (pseudo-)metric on defined by

See [a2] for a history and an exposition of the definite results. This work produced also tools in order to study (non-Gaussian) Banach-valued stochastic processes.

References

[a1]	J. Neveu, "Processus aléatoires Gaussiens" , Univ. Montréal (1968)
[a2]	X. Fernique, "Fonctions aléatores gaussiennes, les résultats de M. Talagrand" Astérisque , 145–146 (1987) pp. 177–186 (Exp. 660, Sém. Bourbaki 1985/86)