Strong Law for the eigenvalues and eigenvectors of empirical covariance matrices

In this paper the following assertion for eigenvalues Ai(jRm n) < · · · < ^mn (-^mn) **d for components Uij of random orthogonal eignevectors {Γχ , . . . , Um of an empirical covariance matrix Rmn of large dimension mn is proved: let 0 < C\ < ^l(Rmn) < ' ' ' < ^mn(^mn) < <?2 < OO be the eigenvalues and \ , . . . , Um be the orthogonal eignevectors of covariance matrix Rmn and the vectors X\ , . . . , Xn be a sample of independent observations of a random vector which are defined on the common probability space, xk = Rliltk + a, Eft = 0, E0t = 7m„ , = { f a , i = 1, . . . , mn}, the components ζα* be independent and for some δ > 0 sup max Ε|ξ,·*| + < oo, lim sup mn < 1. η •f.; — '"' η-*οο Then with probability one and for every p = 1, . . ., m k Jlirn^ fc=max J]£ pi G<>(\k(Rm J) | = 0, where Gp (x) = f£ fp (y) dy and the density fp (y) is equal to mn for all k such that 0 < C$ < km~ < €4 < 1, lim lim min< e|0n-^oo (^ *k(Rm*)-8\ip\x:Fn(x)< _ + e l 1 = 0, l m J U Trans/ated b.v A. 7. V/adimirova

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