Expand the pdf of Wishart distribution into power series via orthogonal polynomials

Question

In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Indeed, since the Laguerre polynomials are related to the derivatives w.r.t. the variance $\sigma^2$, this expansion is exactly the Taylor expansion.

Does anyone know if we can do the same for the general multivariate case, the Wishart distribution? Maybe now we need a matrix variate orthogonal polynomials.

If this is possible, does the expansion still works for the singular case? (i.e., the sample size n is smaller than the dimension p)

Answer 1

Note that the Laguerre orthogonal polynomials are in form of [1](bearing combinatoric interpretation) and [3]

\begin{align} & L_n^\nu(x)=(-1)^n\sum_{m=0}^n \binom n m \prod_{i=1}^m (\nu+2(n-i))(-x)^{n-m} \\[8pt] = {} & \sum_{m=0}^n \frac{\Gamma(\nu+n+1)\frac{\Gamma(-n+m)}{\Gamma(-n)}} {n!m!(\nu+m+1)} x^k=\sum_{m=0}^n c_m(n,\nu) x^k \end{align}

The connection between Wishart matrix and the Laguerre polynomial is that the eigenvalues of the Wishart matrix $M_s = \frac{1}{s} Z_s Z_s^T$ where $Z_s\sim \left[\operatorname{Normal}(0,1)\right]_{n\times s}$ are the zeros of appropriately scaled generalized Laguerre polynomials as explained in [2]. Therefore the spectral density of the Wishart distribution can be written in terms of Laguerre polynomials, and we know that the moments of distribution can be directly derived from its spectral density. Now we obtain relation inbetween moments of Wishart distribution $\beta=1,2,4$ and corresponding Laguerre polynomials in [3] when the Wishart distribution is central. For noncentral case, the derivation in [3] may also apply but with slight modification of the intermediate quantity

$$ \mathcal{Q}(r;m,\ell;\alpha):={ \int_0^\infty dx \, x^r e^{-x} L_m^\alpha (x) L_\ell^\alpha (x)}, $$

which is no longer of the simple form

$$\sum_{k=0,\cdots m,k'=0\cdots\ell} c_k(m,\alpha) c_{k^{'}} (\ell,\alpha)\Gamma(1+r+k+k').$$

Reference

[1]Kuriki, Satoshi, and Yasuhide Numata. "Graph presentations for moments of noncentral Wishart distributions and their applications." Annals of the Institute of Statistical Mathematics 62.4 (2010): 645-672.

This reference is pointed out to us by Carlo.

[2]Dette, Holger. "Strong approximation of eigenvalues of large dimensional Wishart matrices by roots of generalized Laguerre polynomials." Journal of Approximation Theory 118.2 (2002): 290-304.

[3]Livan, Giacomo, and Pierpaolo Vivo. "Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities." arXiv preprint arXiv:1103.2638 (2011).