Guassian Processs

GUASSIAN PROCESSS:

A random process X(t) is a Gaussian process if for all n and all (t1 ,t2 ,…,tn ), the random variables have a jointly Gaussian density function. For Gaussian processes, knowledge of the mean and autocorrelation; i.e., mX (t) and Rx (t1 ,t2 ) gives a complete statistical description of the process. If the Gaussian process X(t) is passed through an LTI system, then the output process Y(t) will also be a Gaussian process. For Gaussian processes, WSS and strict stationary are equivalent.

A Gaussian process is a stochastic process Xt, t ∈ T, for which any finite linear combination of samples has a joint Gaussian distribution. More accurately, any linear functional applied to the sample functionXt will give a normally distributed result.  Notation-wise, one can write X ~ GP(m,K), meaning the random function X is distributed as a GP with mean function m and covariance function K.[1] When the input vector t is two- or multi-dimensional a Gaussian process might be also known as a Gaussian random field.

A sufficient condition for the ergodicity of the stationary zero-mean Gaussian process X(t) is that

 Jointly Gaussian processes:

The random processes X(t) and Y(t) are jointly Gaussian if for all n, m and all (t1 ,t2 ,…,tn ), and (τ1 , τ2 ,…, τm ), the random vector (X(t1 ),X(t2 ),…,X(tn ), Y(τ1 ),Y( τ2 ),…, Y(τm )) is distributed according to an n+M dimensional jointly Gaussian distribution.

For jointly Gaussian processes, uncorrelatedness and independence are equivalent.