an> for <span class="math inline">\(0&lt;\alpha_j
&lt;1, j=1, \ldots, d\)</span>. The spectral density of the
multilogistic distribution is thus <span class="math display">\[\begin{align*}
h_{\boldsymbol{W}}(\boldsymbol{w}) = \frac{1}{d} \left(\sum_{j=1}^d
\alpha_ju_j\right)^{-1} \left(\prod_{j=1}^d \alpha_ju_d
\right)\left(\sum_{j=1}^d
\frac{\Gamma(d-\alpha_j)}{\Gamma(1-\alpha_j)}u_j^{-\alpha_j}\right)\prod_{j=1}^d
w_j^{-1}
\end{align*}\]</span> for <span class="math inline">\(\alpha_j \in
(0,1)\)</span> <