Limiting spectral distribution for a class of random matrices
Let \(X=\{X_{ij}\), \(i,j=1,2,...\}\) be an infinite dimensional random matrix of i.i.d. entries and E \(X^ 2_{11}<\infty\). Let \(T_ p\) be a \(p\times p\) nonnegative definite random matrix independent of X, for \(p=1,2,... \). Suppose \(p^{-1}tr T^ k_ p\to H_ k\) a.s. as \(p\to \infty\) for \(k=1,2,...\), and \(\sum H_{2k}^{-(2k)^{-1}}<\infty\). Then the spectral distribution of \(A_ p=n^{-1}X_ pX_ p'T_ p\), where \(X_ p=\{X_{ij}\), \(i=1,...,p\), \(j=1,...,n\}\) tends to a nonrandom limit distribution as \(p\to \infty\), \(n\to \infty\), but \(p/n\to y>0.\) The two-stage truncation method was used in proving the main result. The notion of M-graphs was introduced and some properties were derived.
existence of limiting spectral distribution
scientific article; zbMATH DE number 3994778