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Strong law of large numbers for functionals of products of random matrices

Authors:

G V Akulov

Institution:

1. Kiev University, USSR

Abstract:

For the sequence A₁, A₂, ..., A_n, ... of m×m independent random matrices such that for each k there exists a joint density function P_k(X) of the elements xgr

_ij ^k, we prove the following theorem: if $sup_{i,j,k} M|\xi _{ij} ^{(k)} |^{\delta _1 }< \infty$ and $sup_i \smallint P_i^{1 + \delta _2 } (X)dX< \infty$ for some positive constants theta

₁ and

₂, then with probability 1,

$\mathop {\lim }\limits_{n \to \infty } n^{ - 1} \{ 1n\|(\mathop \Pi \limits_{i = 1}^n A_i )_{pl} \| - M1n\|(\mathop \Pi \limits_{i = 1}^n A_i )_{pl} \|\} = 0.$

Keywords:
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