Twenty five years of stochastic canonical equation for normalized spectral functions of random symmetric matrices (Q2722263)

The theory of random matrices has developed very quickly. Now it has a great influence on other sciences and has applications in statistics, nuclear physics, linear programming, etc., and the contemporary results of the theory of random matrices are very significant. This paper is devoted to the system of stochastic canonical equations (SCE). The first SCE published by \textit{V. L. Girko} [Random matrices. (Kiev 1975; Zbl 0344.60001)] is a prototype. The author examines this first SCE to provide an introduction to the classical modern technique of the theory of SCE. NEWLINENEWLINENEWLINEIn this paper symmetric matrices \(A_{n}\) with random entries are considered. In the general case entries of the random matrices \(A_{n}=(a_{ij}^{(n)})\) have an arbitrary form and their expectations may not be equal to zero. Under the assumptions, which are not restrictive, that the r.v. \(a_{ij}^{(n)}\) are equal to \(b_{ij}^{(n)}+c_{ij}^{(n)}, i,j,=1,\dots,n,\) where \(b_{ij}^{(n)}\) are nonrandom variables, \(c_{ij}^{(n)},\) are infinitesimal and the vector rows of the random matrices \((b_{ij}+c_{ij})\) are asymptotically constant, it was proved in the previous book that the normalized spectral function of the matrix \((b_{ij}+c_{ij})\) for large \(n\) approaches the normalized spectral function of the matrix \((b_{ij}+d_{ij}),\) where \(d_{ij}, i>j\) are infinitely divisible independent r.v. with given characteristic function. NEWLINENEWLINENEWLINEIn this paper it is shown that it is possible to get a stochastic canonical equation for the Stieltjes transformations of the limit spectral functions of the matrix \((b_{ij}+d_{ij}).\) NEWLINENEWLINENEWLINEThis paper is a good introduction to the history and development of the theory of stochastic canonical equations for normalized spectral functions of random symmetric matrices and contains a survey of results since 1975.