Stochastic canonical equation for normalized spectral functions (Q2722270)

The canonical equation for the Stieltjes transforms of normalized spectral functions of random Gram matrices in the general case is considered when the Lindeberg condition for random entries is not satisfied. It is proved that in this case the normalized traces of resolvents of random Gram matrices \(RR^{T}\) converge to the normalized traces of resolvent of Gram matrices \(GG^{T},\) where instead of random matrices \(R\) we have the sum \(G=A+B\) of nonrandom \(A\) and some diagonal random matrices \(B.\) NEWLINENEWLINENEWLINEThe general formula for the Stieltjes transform of limit normalized spectral function of eigenvalues of the large order random Gram matrices was found in \textit{V. L. Girko} [Random matrices (Kiev 1975; Zbl 0344.60001)] with the help of stochastic canonical equation for random Gram matrices. The paper consists of 13 sections and it is a good introduction into the history and development of the theory of stochastic canonical equations for spectral functions of random Gram matrices.