Joint MGF of ${\bf y}$.

We start with the joint MGF of ${\bf y}$, which is (see [54], p. 68, eq. B-18)

$$\begin{array}{l} g_0(\alpha_1, \alpha_2 \ldots \alpha_N) = \\... ..._1 +\alpha_2 + \cdots + \alpha_N \over N} \right) } \end{array}$$ (8.8)

for ${\rm Re}(\alpha_1)<1$, ${\rm Re}(\alpha_1+\alpha_2)<2$, ... , ${\rm Re}(\alpha_1+\alpha_2+\cdots + \alpha_N)<N$. We can rewrite (8.8) as

$$g_0(\alpha_1, \alpha_2 \ldots \alpha_N) = {1 \over \prod_{n=1}^N \phi_n( \alpha_1,\alpha_2 \ldots \alpha_N)},$$

where

$$\phi_n( \alpha_1,\alpha_2 \ldots \alpha_N) = 1-\frac{1}{n} \sum_{p=1}^n \alpha_p, \;\;\; 1\leq n \leq N.$$

Alternatively,

$$\phi_n( \alpha_1,\alpha_2 \ldots \alpha_N) = 1-\sum_{p=1}^N \; q_{np}\;\alpha_p, \;\;\; 1\leq n \leq N,$$

where

$$q_{np} = \left\{ \begin{array}{lcl} \frac{1}{n}&\mbox{for}&1\leq p\leq n \\ 0&\mbox{for}&n< p\leq N \end{array}\right\}, n=1,2 \ldots N.$$

Define the $N$-by-$N$ matrix ${\bf Q}=[q_{np}]$ and $\alpha$$=[\alpha_1\;\alpha_2 \cdots \alpha_N]^\prime$. Then,

   $$\mbox{\boldmath$\phi$}$$$$($$$$\mbox{\boldmath$\alpha$}$$$$) \stackrel{\mbox{\tiny$\Delta$}}{=} \left[ \begin{array}{c} \p... ... \end{array}\right] = {\underline{\bf 1}}- {\bf Q} \mbox{\boldmath$\alpha$},$$

where ${\underline{\bf 1}}$ is an $N$-by-1 column vector of ones. Thus, $g_0($$\alpha$$)$ is the reciprocal of the product of the elements of $\phi$$($$\alpha$$)$, denoted by

$$g_0( \mbox{\boldmath$\alpha$} ) = {1 \over {\rm prod}({\underline{\bf 1}}- {\bf Q} \mbox{\boldmath$\alpha$})},$$ (8.9)

where ${\rm prod}(\;)$ is the product of the elements of the argument.