Eigenfunctions of Expected Value Operators in the Wishart Distribution

Abstract

Let $(X_{1,m}, X_{2,m}, \cdots X_{k,m}), 1 \leqslant m \leqslant n,$ be a sample of size $n$ from the $k$ dimensional normal distribution with mean vector $\mu$ and covariance matrix $\Sigma.$ Let $V = (\nu_{ij}), 1 \leqslant i, j \leqslant k,$ denote the symmetric scatter matrix where $v_{ij} = \sum_m(X_{i,m} - \mu_i)(X_{j,m} - \mu_j).$ The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar valued functions, $f(V),$ such that $E(f(V)) = \lambda_{n,k}f(\Sigma).$ If $f$ is an eigenfunction then (a) for nonsingular $T,f(T'VT)$ is an eigenfunction and (b) for integral $p, |V|^{p/2}f(V)$ is an eigenfunction. For $k \leqslant 2,$ a complete solution of the problem is given. For $k = 1$ the functions $f(v) = cv^\alpha$ are the only eigenfunctions. For $k = 2,$ a function $f$ is an eigenfunction if and only if (i) $f$ is homogeneous and (ii) $4 \frac{\partial^2 f}{\partial v_{22}\partial v_{11}} - \frac{\partial^2 f}{\partial^2v_{21}} = C|V|^{-1}f.$ A representation of eigenfunctions is given in terms of sums of associated Legendre functions. Relationships between eigenfunctions and harmonic functions are indicated. Any homogeneous polynomial is proved to be a linear combination of polynomial eigenfunctions.