Editing Gaussian process (section)

==Definition==
A time continuous [[stochastic process]] <math>\left\{X_t ; t\in T\right\}</math> is Gaussian [[if and only if]] for every [[finite set]] of [[indexed family|indices]] <math>t_1,\ldots,t_k</math> in the index set <math>T</math>

<math display="block"> \mathbf{X}_{t_1, \ldots, t_k} = (X_{t_1}, \ldots, X_{t_k}) </math>

is a [[multivariate normal distribution|multivariate Gaussian]] [[random variable]].<ref name="DrMacKayGPNN">{{cite book| last=MacKay | first=David J.C.| author-link=David J.C. MacKay| title=Information Theory, Inference, and Learning Algorithms |year=2003 | pages=540|publisher=[[Cambridge University Press]]| isbn=9780521642989|url=http://www.inference.phy.cam.ac.uk/itprnn/book.pdf | quote=The probability distribution of a function <math>y(\mathbf{x})</math> is a Gaussian processes if for any finite selection of points <math>\mathbf{x}^{(1)},\mathbf{x}^{(2)},\ldots,\mathbf{x}^{(N)}</math>, the density <math>P(y(\mathbf{x}^{(1)}),y(\mathbf{x}^{(2)}),\ldots,y(\mathbf{x}^{(N)}))</math> is a Gaussian}}</ref> 
As the sum of independent and Gaussian distributed random variables is again Gaussian distributed, that is the same as saying every linear combination of <math> (X_{t_1}, \ldots, X_{t_k}) </math> has a univariate Gaussian (or normal) distribution.

Using [[Characteristic function (probability theory)|characteristic functions]] of random variables with <math>i</math> denoting the [[imaginary unit]] such that <math>i^2 =-1</math>, the Gaussian property can be formulated as follows: <math>\left\{X_t ; t\in T\right\}</math> is Gaussian if and only if, for every finite set of indices <math>t_1,\ldots,t_k</math>, there are real-valued <math>\sigma_{\ell j}</math>, <math>\mu_\ell</math> with <math>\sigma_{jj} > 0</math> such that the following equality holds for all <math>s_1,s_2,\ldots,s_k\in\mathbb{R}</math>,

<math display="block"> { \mathbb E }\left[\exp\left(i \sum_{\ell=1}^k s_\ell \, \mathbf{X}_{t_\ell}\right)\right] = \exp \left(-\tfrac{1}{2} \sum_{\ell, j} \sigma_{\ell j} s_\ell s_j + i \sum_\ell \mu_\ell s_\ell\right),</math>

or <math> { \mathbb E } \left[ {\mathrm e}^{ i\, \mathbf{s}\, (\mathbf{X}_t - \mathbf{\mu}) } \right] = {\mathrm e}^{  - \mathbf{s}\, \sigma\, \mathbf{s}/2 }</math>. 
The numbers <math>\sigma_{\ell j}</math> and <math>\mu_\ell</math> can be shown to be the [[covariance]]s and [[mean (mathematics)|means]] of the variables in the process.<ref>{{cite book |last=Dudley |first=R.M. |title=Real Analysis and Probability |year=1989 |publisher=Wadsworth and Brooks/Cole |isbn=0-534-10050-3 }}</ref>