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# The Gaussian Process

Module by: Tuan Do-Hong. E-mail the author

## Gaussian Random Processes

Definition 1: Gaussian process
A process with mean μ X t μ X t and covariance function C X t 2 t 1 C X t 2 t 1 is said to be a Gaussian process if any X= X t 1 X t 2 X t N T X X t 1 X t 2 X t N formed by any sampling of the process is a Gaussian random vector, that is,
f X x=12πN2det Σ X 12e(12(xμX)T Σ X -1(xμX)) f X x 1 2 N 2 Σ X 1 2 1 2 x μ X Σ X x μ X
(1)
for all xRn x n where μX= μ X t 1 μ X t N μ X μ X t 1 μ X t N and Σ X =( C X t 1 t 1 C X t 1 t N C X t N t 1 C X t N t N ) Σ X C X t 1 t 1 C X t 1 t N C X t N t 1 C X t N t N . The complete statistical properties of X t X t can be obtained from the second-order statistics.

### Properties

1. If a Gaussian process is WSS, then it is strictly stationary.
2. If two Gaussian processes are uncorrelated, then they are also statistically independent.
3. Any linear processing of a Gaussian process results in a Gaussian process.

### Example 1

XX and YY are Gaussian and zero mean and independent. Z=X+Y Z X Y is also Gaussian.

φ X u=eiuX-=e(u22 σ X 2 ) φ X u u X u 2 2 σ X 2
(2)
for all uR u
φ Z u=eiu(X+Y)-=e(u22 σ X 2 )e(u22 σ Y 2 )=e(u22( σ X 2 + σ Y 2 )) φ Z u u X Y u 2 2 σ X 2 u 2 2 σ Y 2 u 2 2 σ X 2 σ Y 2
(3)
therefore ZZ is also Gaussian.

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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