Thirty years of the ACE-Law and Stochastic Power Method

This paper is devoted to the very old problem of describing all possible limit distributions functions of eigenvalues and the components of eigenvectors of the Girko's ensemble of random matrices with independent pairs of asymptotically constant entries(ACE) (ξ_ij, ξ_ij), i ≥ j. In spite of such a general formulation of the problem, we can surprisingly solve it, and the solution is very simple and it is easy to remember the ACE-law: under some wide conditions the possible limit distributions for random eigenvalues of random matrices Ξ_n = (ξ_ij) coincide with corresponding distributions of the members of order statistics built by entries ξ_ij of random matrices. The reason of publication of this survey consists in recurring recent publications which have borrowed my results without saying "merci." Therefore I consider these publications to be some kind of unfunny joke. Although my results were published 30 years ago and were translated into English, there are no references to my work in that papers. I do believe that there is no such thing as intellectual property on mathematical results. My results do not belong to me, they belong to the mankind. Yet, it strikes me as particularly crude that some particular cases of the formulas (27.4) and (44.5) has recently been published by certain people, without making due reference to my publications [5], which dates back as far as 1973.

We give here all ACE_k, k = 1, ..., 55-laws named according to the abbreviation of Asymptotical Constant Entries of random matrices. It is easy to remember my ACE-Law: the limit distribution of any members of ordered in increasing order eigenvalues converges when n → ∞ with corresponding distributions of the members of order statistics of some function of random entries of random matrices. My statements are very simple, moreover the proofs also are very simple and are easily understood by students of mathematical departments, and is based on the VICTORIA-transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays.