Twenty Five Years of Stochastic Canonical Equation K ₃₉ for Normalized Spectral Functions of Ace-Symmetric Matrices

Abstract

To provide an introduction to classical modern techniques of the theory of stochastic canonical equations, we examine the first stochastic canonical equation, which was found in [Gir12] twenty five years ago. In this chapter we will consider symmetric matrices Ξ _n with random entries. In the general case, the entries of the random matrix ${\Xi _n} = (\xi _{ij}^{(n)})$ have an arbitrary form and their expectations may not be equal to zero. If the entries $\xi _{ij}^{(n)},\;i \ge j,\;\;i,j = 1,\; \ldots ,\;n$ of the symmetric random matrix Ξ _n are independent, then it follows then from the main theorem of the theory of random matrices that the normalized spectral function µ _n (x) of such a matrix can be approximately replaced by the nonrandom spectral function Eµ _n (x) (see [Gir12, pp. 217–219]). The main problem is to find all limit of spectral function Eµ _n (x). Without any additional assumptions for random variables $\xi _{ij}^{(n)}$ this problem seems trivial, since any distribution function can be a limiting function. We shall make the following assumptions, which are not too restrictive and are confirmed by numerous problems: the random variables $\xi _{ij}^{(n)}$ are equal to asymptotically constant entries $a_{ij}^{(n)} + \nu _{ij}^{(n)},\;i, j = 1, \ldots , n, where a_{ij}^{(n)}$ are nonrandom variables, $\nu _{ij}^{(n)},\;i, j = 1, \ldots , n$ are infinitesimal and the vector rows of the random matrices (a _ij + v _ij ) are asymptotically constant. It was proved in [Gir12, p.241] that the normalized spectral function of the ACE-matrix (a _ij + v _ij ) for large n approaches the normalized spectral function of the matrix (a _ij + γ _ij ) where γ _ij , i > j, are infinitely divisible independent random variables with the characteristic functions exp {E exp(isv _ij ) - 1}. In this chapter, we show that it is possible to get a canonical equation for the Stieltjes transformations of the limit spectral functions of the matrix (a _ij + γ _ij ).