V. L. Girko
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) (ξij
, ξij
), 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 Ξn = (ξij
) coincide with corresponding distributions of
the members of order statistics built by entries ξij
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 ACEk, 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.
Random Operators and Stochastic Equations, Walter de Gruyter
Print ISSN: 0926-6364
Volume: 13, 06/2005
Pages: 153 - 206
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