Fundamental Multidimensional Variables

Abstract

The joint PDF of a Gaussian vector ${\rm X} \in N_n \left( {\bar {\rm X},\sigma ^2 } \right)$ with covariance matrix

$$ {\rm M}_X = E\left\{ {\left( {{\rm X} - \overline {\rm X} } \right)\left( {{\rm X} - \overline {\rm X} } \right)^t } \right\} $$

(3.1)

is given as

$$ p_X \left( x \right) = \frac{1} {{\left( {2\pi } \right)^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \left| {M_X } \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}\exp \left[ { - \frac{1} {2}\left( {x - \overline {\rm X} } \right)^t M_{_X }^{ - 1} \left( {x - \overline {\rm X} } \right)} \right] $$

(3.2)

where |M_X| denotes the determinant of M_X. Letting

$$ \omega = \left[ \begin{gathered} \omega _1 \\ \omega _1 \\ \cdot \\ \cdot \\ \omega _n \\ \end{gathered} \right] $$

(3.3)

then the joint CF is given by

$$ \Psi _X \left( \omega \right) = \exp \left[ {j\omega ^t \overline {\rm X} - \frac{1} {2}\omega ^t {\rm M}_X \omega } \right] $$

(3.4)

For the special case of n = 2, $\bar {\rm X} = 0$, and the covariance matrix of (1.10), the joint moments are given as

$$ E\left\{ {X_1^{k_1 } X_2^{k_2 } } \right\} = \left\{ \begin{gathered} 0,k_1 + k_2 odd \hfill \\ \sigma ^{k_1 + k_2 } \sum\limits_{i = 0}^{\left\lfloor {{{k_1 } \mathord{\left/ {\vphantom {{k_1 } 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rfloor } {\left( \begin{gathered} k_1 \hfill \\ 2i \hfill \\ \end{gathered} \right)\left( {1 - \rho ^2 } \right)^i \rho ^{k_1 - 2i} \left( {k_1 + k_2 - 2i - 1} \right)} !!\left( {2i - 1} \right)!!, \hfill \\ k_1 + k_2 even \hfill \\ \end{gathered} \right. $$

(3.5)

.