Canonical Equation K ₆ for Symmetric Random Matrices with Identically Distributed Entries

Abstract

Random matrices with identically distributed entries always draw the attention of experts in the field of probability theory and its applications. It is well known that if the variances of these entries for symmetric random matrices exist, then we have the so-called Wigner semicircle law. But what happens if absolute moments of the entries of random matrices do not exist? For the sum of independent identically distributed random variables the answer is well known: all possible limit distributions belong to the so-called class of stable laws. Unfortunately, we still cannot describe all possible limited normalized spectral functions for matrices with identically distributed entries, but if we assume that the distribution of random entries belongs to the region of attraction of a stable law with parameter 0 < α ≤ 2, then we can find the Stieltjes transform of the limit spectral function satisfying a certain stable stochastic equation.