# Gaussian process

2010 Mathematics Subject Classification: Primary: 60G15 [MSN][ZBL]

A real stochastic process $X = X( t)$, $t \in T$, all finite-dimensional distributions of which are Gaussian, i.e. for any $t _ {1} \dots t _ {n} \in T$ the characteristic function of the joint probability distribution of the random variables $X( t _ {1} ) \dots X( t _ {n} )$ has the form

$$\phi _ {t _ {1} \dots t _ {n} } ( u _ {1} \dots u _ {n} ) =$$

$$= \ \mathop{\rm exp} \left \{ i \sum _ {k = 1 } ^ { n } A ( t _ {k} ) u _ {k} - { \frac{1}{2} } \sum _ {k, j = 1 } ^ { n } B ( t _ {k} , t _ {j} ) u _ {k} u _ {j} \right \} ,$$

where $A( t) = {\mathsf E} X( t)$ is the mathematical expectation and

$$B ( t, s) = {\mathsf E} [ X ( t) - A ( t)] [ X ( s) - A ( s)]$$

is the covariance function. The probability distribution $X = X( t)$ of a Gaussian process is completely determined by its mathematical expectation $A( t)$ and by the covariance function $B( t, s)$, $s, t \in T$. For any function $A( t)$ and any positive-definite function $B( t, s)$ there exists a Gaussian process $X( t)$ with expectation $A( t)$ and covariance function $B( t, s)$. A multi-dimensional stochastic process with vector values

$$X ( t) = \{ X _ {1} ( t) \dots X _ {m} ( t) \}$$

is called Gaussian if the joint probability distributions of arbitrary variables

$$X _ {i _ {1} } ( t _ {1} ) \dots X _ {i _ {n} } ( t _ {n} )$$

are Gaussian.

A complex Gaussian process $X = X( t)$, $t \in T$, is a process of the form

$$X ( t) = X _ {1} ( t) + iX _ {2} ( t),$$

in which $X _ {1} ( t)$, $X _ {2} ( t)$ jointly form a two-dimensional real Gaussian process. Regarding a complex Gaussian process $X( t) = X _ {1} ( t) + i X _ {2} ( t)$ one additional stipulation is imposed:

$${\mathsf E} X ( s) X ( t) = A ( s) A ( t),$$

where

$$A ( t) = {\mathsf E} X ( t).$$

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:

$${\mathsf E} [ X _ {1} ( t) - A _ {1} ( t)] [ X _ {1} ( s) - A _ {1} ( s)] =$$

$$= \ {\mathsf E} [ X _ {2} ( t) - A _ {2} ( t)] [ X _ {2} ( s) - A _ {2} ( s)] = { \frac{1}{2} } \mathop{\rm Re} B ( t, s),$$

$${\mathsf E} [ X _ {1} ( t) - A _ {1} ( t)] [ X _ {2} ( s) - A _ {2} ( s) ] = - { \frac{1}{2} } \mathop{\rm Im} B ( t, s),$$

where

$$B ( t, s) = {\mathsf E} [ X ( t) - A ( t)] \overline{ {[ X ( s) - A ( s)] }}\;$$

is the covariance function of the process $X( t)$ and

$$A _ {1} ( t) = {\mathsf E} X _ {1} ( t),\ \ A _ {2} ( t) = {\mathsf E} X _ {2} ( t).$$

A linear generalized stochastic process $X = \langle u , X \rangle$, $u \in U$, on a linear space $U$ is called a generalized Gaussian process if its characteristic functional $\phi _ {X} ( u )$ has the form

$$\phi _ {X} ( u) = e ^ {iA ( u) - B ( u, u) /2 } ,\ \ u \in U ,$$

where $A( u ) = {\mathsf E} \langle u , X\rangle$ is the mathematical expectation of the generalized process $X = \langle u , X\rangle$ and

$$B ( u , v) = \ {\mathsf E} [ \langle u , X\rangle - A ( u)] [ \langle v, X\rangle - A ( v)]$$

is its covariance functional.

Let $U$ be a Hilbert space with scalar product $( u , v)$, $u , v \in U$. A random variable $X$ with values in $U$ is called Gaussian if $X = \langle u , X\rangle$, $u \in U$, is a generalized Gaussian process. The mathematical expectation $A( u)$ is a continuous linear functional, while the covariance function $B( u , v)$ is a continuous bilinear functional on the Hilbert space $U$, and

$$B ( u , v) = ( Bu , v),\ \ u , v \in U,$$

where the positive operator $B$ is a nuclear operator, called the covariance operator. For any such $A( u )$ and $B( u , v)$ there exists a Gaussian variable $X \in U$ such that the generalized process $X = \langle u , X\rangle$, $u \in U$, has expectation $A( u )$ and covariance function $B( u , v)$.

Example. Let $X = X( t)$ be a Gaussian process on the segment $T = [ a, b]$, let the process $X( t)$ be measurable, and let also

$$\int\limits _ { a } ^ { b } {\mathsf E} [ X ( t)] ^ {2} dt < \infty .$$

Then almost-all the trajectories of $X( t)$, $t \in T$, will belong to the space of square-integrable functions $u = u( t)$ on $T$ with the scalar product

$$( u , v) = \ \int\limits _ { a } ^ { b } u ( t) v ( t) dt.$$

The formula

$$\langle u , X\rangle = \ \int\limits _ { a } ^ { b } u ( t) X ( t) dt,\ \ u \in U,$$

defines a generalized Gaussian process on this space $U$. The expectation and the covariance functional of the generalized process $X = \langle u , X\rangle$ are expressed by the formulas

$$A ( u) = \int\limits _ { a } ^ { b } u ( t) A ( t) dt,$$

$$B ( u , v) = \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } B ( t, s) u ( t) v ( s) dt ds,$$

where $A( t)$ and $B( t, s)$ are, respectively, the expectation and the covariance function of the initial process $X = X( t)$ on $T = [ a, b]$.

Almost-all the fundamental properties of a Gaussian process $X = X( t)$( the parameter $t$ runs through an arbitrary set $T$) may be expressed in geometrical terms if the process is considered as a curve in the Hilbert space $H$ of all random variables $Y$, ${\mathsf E} Y ^ {2} < \infty$, with the scalar product $( Y _ {1} , Y _ {2} ) = {\mathsf E} Y _ {1} Y _ {2}$ for which

$$( X ( t), 1) = A ( t),$$

and

$$( X ( t) - A ( t), X ( s) - A ( s)) = B ( t, s).$$

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 [D]). 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 $H$ introduced in [I], [I2]. The first actual examples of dynamical systems with "non-classical" spectral properties have been constructed in this way.

How to Cite This Entry:
Gaussian process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gaussian_process&oldid=47056
This article was adapted from an original article by Yu.A. Rozanov, D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article