The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices.

One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for instance, Muirhead: "Aspects of Multivariate Analysis") the integral

\begin{equation} \phi(\Theta) = \int_{P_p} \exp(\sum_{j\le k}^p \theta_{jk} a_{jk}) f(A)\; dA \end{equation}

where $f(A)$ is the density function of $A$. (And $\Theta$ is a symmetric $p\times p$-matrix)

So the question is: I am searching for references for this Laplace transform, inversion theorems, numerical methods, known transform formulas, .... etc ???

The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices.

One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for instance, Muirhead: "Aspects of Multivariate Analysis") the integral

\begin{equation} \phi(\Theta) = \int_{P_p} \exp(\sum_{j\le k}^p \theta_{jk} a_{jk}) f(A)\; dA \end{equation}

where $f(A)$ is the density function of $A$. (And $\Theta$ is a symmetric $p\times p$-matrix)

So the question is: I am searching for references for this Laplace transform, inversion theorems, numerical methods, known transform formulas, .... etc ???

Thanks for answers! Now, I am searching for those references, but one is really difficult to find, that is, volume 2 of Audrey Terras' book: "Harmonic analysis on symmetric spaces II"

I have found the first volume, but volume II cannot even be found on Springers own website! Any ideas about how to find it?