V. A. MARCENKO AND L. A. PASTURdistribution of this random quantity is one of the fundamental problems in the spectral analysis of random operators. Of particular interest is the case of very large Ν and n, since it often appears that for Ν -> °o the random quantity uiX; Β ρ/ (η)) converges in probability to a nonrandom number.We assume the following conditions are satisfied for Ν -> <χ>.I. The limit litnp/^mn/N = c, which for brevity we call the concentration, exists.II. The sequence of normalized spectral functions vi\; Ap/) of the operators Ap/ converges to some function v o (\) at all points of continuity: lim ν (λ;Α Ν ) = v 0 (λ).(1-2)Assuming that these conditions are satisfied, it is necessary, first of all, to make clear how the stochastic properties of operators B N {n) of the form in (1-1) ensure the convergence in probability of the sequences ν {λ; Βpj (n)) to nonrandom numbers, i.e. to explain when a nondecreasing function !/(λ; c) exists such that at all of its points of continuity

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We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X * X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, ∞) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5 ). This is done by using a simple "interpolation trick" from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially C 5 test function. Here the variance of statistics contains an additional term proportional to κ4. The proofs of all limit theorems follow essentially the same scheme.