Personal tools
•
You are here: Home Random Vectors

Random Vectors

Document Actions

Vectors  ::  Covariance  ::  Functions  ::  Application  ::  Markov Model

Characteristic functions

As before, this is just an -dimensional Fourier transform.

Properties of Gaussian random vectors:

1. .
2. independent if and only if is a diagonal matrix.
3. If , then is also Gaussian,

Linear functions of Gaussians are Gaussians.

Said another way: Family of Gaussians closed under affine transformations.

Suppose is positive definite . Then it can be factored as

where is an invertible, lower-triangular matrix. This factorization is called the Cholesky factorization . This is essentially a matrix square root.''

Suppose , with p.d. Let . Then is normal with and .

This process of diagonalizing the covariance matrix is called whitening. We say that uncorrelated i.i.d. components are white .

4. If (i.e., p.d.) then is a continuous r.v. with

where product of eigenvalues.
5. Important: Suppose with . Partition ,

where has elements. It turns out that is also Gaussian. (How could we easily show this?) Let us partition

Then

Consider conditioned on :

Then it can be shown that

where

This is smaller'' than .

Discuss implications. Draw pictures.

Note: For a Gaussian vector, the conditional density is Gaussian.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Random Vectors. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec3_3.html. This work is licensed under a Creative Commons License