Spectral analysis of stochastic recurrence systems of growing dimension under G-condition. Canonical equation K 91

Abstract

The spectral analysis of random recurrence systems of growing dimension is considered, when the number of entries (parameters or coefficients) of matrices and the number of their observations have the same order. We develop a random matrix language for a problem of finding the limit spectra of the product of random matrices (matriciant) on the basis of axioms of general statistical analysis (GSA), V.I.C.T.O.R.I.A.-function (Very Important Calculative Transformation of Randomly Independent (Invisible, etc.) Arrays), REFORM method (Resolvents Formulas for Martingales), etc.

Spectral relations between products and powers of isotropic random matrices

“…[20]. Our result elucidates the transparent analytic structure noted in several papers on the products of random matrices [6][7][8][9]12,[21][22][23][24][28][29][30][31][32][33] and provides a powerful tool for the derivation of similar results for products of some application-designed isotropic random matrices of large (infinite) size. and 1 otherwise.…”

Products of Independent non-Hermitian Random Matrices

“…We are grateful to Z. Burda, T. Tao and A. Tikhomirov for useful comments regarding the results of the paper. In addition, we are grateful to unanimous referees for valuable and constructive criticism regarding the proofs of Theorem 15 and Lemma 19, and for bringing to our attention the reference [13] where a similar result was obtained for m = 2.…”

L.I.F.E.: and Halloween Law

“…(This assertion is a simple corollary from equation K 91 which we have found in [3]. ) We denote this result using the symbol " k n B L:I:F:E: Q k j D1 "…”

“…For the first time certain Hermitian analytical functions of powers of matrices " n from the class G 1 were investigated by Wegmann [6]. In our case when a function of the matrix " n is non-Hermitian and belongs to the class G 2 or G 3 Wegmann's method is not valid. Nevertheless, we can find some relatively simple relation for the spectra of analytical functions of random matrices using the L.I.F.E.…”

“…The proof of the following theorem is almost the same as in [3]. Only we should underline that for independent random matrices and for matrices which coincide the corresponding main formulas contain two terms for block matrices: EB pl B T pl and EOE .n/ pl 2 P m P T m where B pl D OE .n;j / pl C " n .n;j / pl p pl p;lD1;:::;j and P m D OEp pl p;lD1;:::;j a certain permutation matrix, see the next section for a detail.…”