Bounds of integration for the Wishart density

Question

I once took a course that included zillions of exercises concerning the Wishart distribution, but as far as I recall, never mentioned the Wishart density. I asked something about that in this question, in which I mentioned that bounds of integration might be messy.

A reply said the measure with respect to which one integrates (in the non-singular case) is $$ \prod_{i,j\,:\,1\,\le\,i\,\le\,j\,\le\,p} dx_{i,j}. $$

My question here is about those "messy" bounds of integration.

I'm going to make a guess and see whether someone who knows something can confirm it or deny it or (maybe the best option) improve upon it.

We are integrating over the space of matrices $(x_{i,j})_{i,j \, \in\, \{1,\,\ldots,\,p\}}$ that are symmetric and positive-definite.

Suppose such a matrix is $$ \left[ \begin{array}{cc} \underset{p_1\times p_1} A & \underset{p_1\times p_2} B \\[8pt] \underset{p_2\times p_1}{B^T} & \underset{p_2\times p_2}C \end{array} \right]. $$

Then $A$ and $C$ are symmetric and positive-definite, and $B$ must be such as to make the entire matrix above positive-definite.

There is this one-to-one correspondence: $$ \left[ \begin{array}{cc} A & B \\[8pt] B^T & C \end{array} \right] \longleftrightarrow \left( \underset{p_1\times p_1} A,\quad \underset{p_2\times p_1} {B^T A^{-1}}, \quad \underset{p_2\times p_2} {C- B^T A^{-1} B} \right) = (J,K,L). $$ So $J$ and $L$ are positive-definite.

The second component of this triple occurs in an expression for conditional expected value and the third is the corresponding conditional variance.

It follows that \begin{align} A & = J, \\ B & = JK^T, \\ C & = L + KJK^T. \end{align} The domain of $(J,K,L)\mapsto(A,B,C)$ is a Cartesian product whose first and third factors are the sets of all positive-definite symmetric real matrices of the appropriate sizes and whose second factor is the set of (here is the interesting part) all $p_2\times p_1$ real matrices $K.$ (Easy exercise: Prove that.)

So we have reduced the problem of bounds of integration to that of bounds of integration for smaller positive-definite symmetric real matrices, plus integrating over a space of matrices in which the bounds for every entry are $-\infty$ and $+\infty.$

This can be iterated until we have a Cartesian product of $p=p_1+p_2$ factors each of which is $(0,+\infty),$ and $\binom p2$ factors each of which is $(-\infty,+\infty).$

My questions are: Is this useful? Is this a standard technique?

Answer 1

Here is the requested theorem

You would probably be interested in the proof, too. It runs over a few pages.

Multiplying differentials is done in the skew-symmetric sense, not in the typical commutative way. This "wedge" product is described in the previous chapter of the book.

Answer 2

I have the most wonderful proof of this sublime theorem, but it is too long to be written in this narrow margin.

Or, more precisely, I have this reduction of the problem of integrating over the space of $3\times3$ positive-definite symmetric real matrices to the problem of integrating over the space of $2\times2$ symmetric real matrices: $$ \int\limits_{\mathbf S^3_{++}} f\left( \left[ \begin{array}{ccc} x_{1,1} & x_{1,2} & x_{1,3} \\ x_{1,2} & x_{2,2} & x_{2,3} \\ x_{1,3} & x_{2,3} & x_{3,3} \end{array} \right] \right) \, d(x_{1,1}, x_{1,2}, x_{1,3}, x_{2,2}, x_{2,3}, x_{3,3}) $$ $$ = \int\limits_{(0,+\infty)} \left(\,\,\, \int\limits_{\mathbb R^2} \left( \,\,\, \int\limits_{\mathbf S^2_{++}} f\left( \left[ \begin{array}{ccc} t & tu_1 & tu_2 \\ tu_1 & v_{1,1} + tu_1^2 & v_{1,2} + tu_1u_2 \\ tu_2 & v_{1,2} + tu_1u_2 & v_{2,2} + tu_2^2 \end{array} \right] \right) \, d(v_{1,1}, v_{1,2}, v_{2,2} ) \right) t^2 \, d(u_1,u_2) \right) \, dt $$ I was going to post this as a comment but it wouldn't fit. The margin was too narrow.