# The condition for random positive matrice integration

For a $$k \times k$$ positive matrix $$V=(v_{ij}),$$ write $$V=\Gamma'D\Gamma,$$ where $$D=diag(d_1,d_2,\ldots,d_k)$$ with $$d_1>d_2>\cdots>d_k,$$ and $$\Gamma$$ is orthogonal matrix. From the result of Farrell (1985), $$dV=\prod_{i here $$dV=\prod_{i\le j} dv_{ij}$$, $$dD=\prod_{i=1}^k dd_{i},$$ and $$d\Gamma$$ denotes over the space of orthonormal matrices.

Question: What is the sufficient and necessary conditions of a,b,c for the following integration is integrable? $$\int\int\frac{1}{|V|^a|W|^b|I_k+V+W|^c\prod_{i where positive matrix $$W=O'\Lambda O,$$ $$\Lambda=diag(\lambda_1,\lambda_2,\ldots,\lambda_k)$$ with $$\lambda_1>\lambda_2>\cdots>\lambda_k,$$ and $$O$$ is orthogonal matrix.

From (1), it follows that $$\int\int\frac{1}{|D|^a|\Lambda|^b|I_k+\Gamma'D\Gamma+O'\Lambda O|^c} dDd\Lambda d\Gamma dO.$$