Stochastic Matrices

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Notation

In all the expressions below, x is a vector of real or complex random variables with whose mean vector and covariance matrix are given by: E(x) = m and Cov(x)=E((x-m)(x-m)H) = S. We define the real-valued vector of variances s=diag(S).

 •,  ÷, •2, ½ and exp() are elementwise operators for multiplication, division, square, square root and exponentiation. ¤ denotes the Kroneker product and j dentotes (-1)½.

Vectors and matrices a, A, b, B, c, C, d and D are constant (i.e. not dependent on x).

General Properties

Special Distributions

The expressions for cubic and quartic expectations given below are restricted to the following special distributions:

Independent

Real Gaussian

Complex Gaussian

Definition: In this section, <=> represents the  Complex-to-Real isomporphism and  <-> represents the related vector mapping.

[x[n]:Complex Gaussian] means that if x[n] <=> y[2n] , then y ~ N(y ; a, ½K) for some complex m[n] <-> a[2n] and +ve definite hermitian S[n#n] <=> K[2n#2n]. In other words, the real and imaginary components of x are jointly gaussian with a symmetric covariance matrix that lies in the range of the complex-to-real isomorphism.

In the following sections, we define  d=s½ to be a positive real-valued vector of standard deviations. The function 1F1(a,b;z)=M(a,b,z) is the Confluent Hypergeometric or Kummer function, hypergeom(a,b,z) in MATLAB. The function 2F1(a,b;z) is the Confluent Hypergeometric function, hypergeom(a,b,z) in MATLAB.

Linear Expectations

Quadratic Expectations

For [x:Real Gaussian] :

For [x:Complex Gaussian] :

Cubic Expectations

For [x:Real Independent] :

For [x:Real Gaussian] :

Quartic Expectations

For [x:Independent] :

For [x:Real Gaussian] :

High Powers

For [x:Real Gaussian] :


This page is part of The Matrix Reference Manual. Copyright © 1998-2005 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "imperial.ac.uk".
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