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.

**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.

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Communication Theory : Random Process : Guassian Processs |

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