View source for Gaussian process

{{MSC|60G15}}

[[Category:Stochastic processes]]

A real stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436001.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436002.png" />, all finite-dimensional distributions of which are Gaussian, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436003.png" /> the characteristic function of the joint probability distribution of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436004.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436005.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436006.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436007.png" /> is the mathematical expectation and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436008.png" /></td> </tr></table>

is the [[Covariance|covariance]] function. The probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g0436009.png" /> of a Gaussian process is completely determined by its mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360010.png" /> and by the covariance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360012.png" />. For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360013.png" /> and any positive-definite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360014.png" /> there exists a Gaussian process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360015.png" /> with expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360016.png" /> and covariance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360017.png" />. A multi-dimensional stochastic process with vector values

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360018.png" /></td> </tr></table>

is called Gaussian if the joint probability distributions of arbitrary variables

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360019.png" /></td> </tr></table>

are Gaussian.

A complex Gaussian process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360021.png" />, is a process of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360022.png" /></td> </tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360024.png" /> jointly form a two-dimensional real Gaussian process. Regarding a complex Gaussian process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360025.png" /> one additional stipulation is imposed:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360026.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360027.png" /></td> </tr></table>

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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360028.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360029.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360030.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360031.png" /></td> </tr></table>

is the covariance function of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360032.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360033.png" /></td> </tr></table>

A linear generalized stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360035.png" />, on a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360036.png" /> is called a generalized Gaussian process if its characteristic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360037.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360038.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360039.png" /> is the mathematical expectation of the generalized process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360040.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360041.png" /></td> </tr></table>

is its covariance functional.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360042.png" /> be a Hilbert space with scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360044.png" />. A random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360045.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360046.png" /> is called Gaussian if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360048.png" />, is a generalized Gaussian process. The mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360049.png" /> is a continuous linear functional, while the covariance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360050.png" /> is a continuous bilinear functional on the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360051.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360052.png" /></td> </tr></table>

where the positive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360053.png" /> is a nuclear operator, called the covariance operator. For any such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360055.png" /> there exists a Gaussian variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360056.png" /> such that the generalized process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360058.png" />, has expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360059.png" /> and covariance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360060.png" />.

Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360061.png" /> be a Gaussian process on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360062.png" />, let the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360063.png" /> be measurable, and let also

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360064.png" /></td> </tr></table>

Then almost-all the trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360066.png" />, will belong to the space of square-integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360068.png" /> with the scalar product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360069.png" /></td> </tr></table>

The formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360070.png" /></td> </tr></table>

defines a generalized Gaussian process on this space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360071.png" />. The expectation and the covariance functional of the generalized process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360072.png" /> are expressed by the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360073.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360074.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360076.png" /> are, respectively, the expectation and the covariance function of the initial process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360077.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360078.png" />.

Almost-all the fundamental properties of a Gaussian process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360079.png" /> (the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360080.png" /> runs through an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360081.png" />) may be expressed in geometrical terms if the process is considered as a curve in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360082.png" /> of all random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360084.png" />, with the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360085.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360086.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360087.png" /></td> </tr></table>

''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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360088.png" /> introduced in [[#References|[4]]], [[#References|[5]]]. The first actual examples of dynamical systems with "non-classical" spectral properties have been constructed in this way.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Doob, "Stochastic processes" , Chapman &amp; Hall (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.A. Ibragimov, Yu.A. Rozanov, "Gaussian random processes" , Springer (1978) (Translated from Russian) {{MR|0543837}} {{ZBL|0392.60037}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Cramér, M.R. Leadbetter, "Stationary and related stochastic processes" , Wiley (1967) pp. Chapts. 33–34 {{MR|0217860}} {{ZBL|0162.21102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> K. Itô, "Multiple Wiener integral" ''J. Math. Soc. Japan'' , '''3''' : 1 (1951) pp. 157–169 {{MR|0044064}} {{ZBL|0044.12202}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Itô, "Complex multiple Wiener integral" ''Japan J. Math.'' , '''22''' (1952) pp. 63–86 {{MR|0063609}} {{ZBL|0049.08602}} </TD></TR></table>

''D.V. Anosov''

====Comments====
A Gaussian process is sometimes called a normal process. See [[Stationary stochastic process|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360089.png" /> with respect to the (pseudo-)metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360090.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360091.png" /> defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043600/g04360092.png" /></td> </tr></table>

See [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Neveu, "Processus aléatoires Gaussiens" , Univ. Montréal (1968) {{MR|0272042}} {{ZBL|0192.54701}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> X. Fernique, "Fonctions aléatores gaussiennes, les résultats de M. Talagrand" ''Astérisque'' , '''145–146''' (1987) pp. 177–186 (Exp. 660, Sém. Bourbaki 1985/86)</TD></TR></table>