A limit theorem for the eigenvalues of product of two random matrices
If \(A_ p\) is a \(p\times p\) matrix with real eigenvalues, and \(pF_ p(x)\) is the number of eigenvalues less than or equal to x, then we call \(F_ p(x)\) the spectral distribution function of \(A_ p\). Let \(W_ p=X_ pX^ T_ p\) be a Wishart matrix where \(X_ p\) is \(p\times m\) dimensional. Let \(T_ p\) be a \(p\times p\) symmetric matrix of random variables. Under certain conditions the authors prove that if \(F_ p\) is the spectral distribution function of \(m^{-1}W_ pT_ p\), then there exists a distribution function F such that \(F_ p(x)\) converges in probability to \(F(x)\), for any x, as p increases without limit.
scientific article; zbMATH DE number 3881667