We study the eigenvalues of large perturbed matrices. We consider a pattern matrix P, we blow it up to get a large block-matrix Bn. We can observe only a noisy version of matrix Bn. So we add a random noise Wn to obtain the perturbed matrix An = Bn + Wn. Our aim is to find the structural eigenvalues of An. We prove asymptotic theorems on this problem and also suggest a graphical method to distinguish the structural and the non-structural eigenvalues of An.

We study the Bahr-Esseen inequality. We show that the Bahr-Esseen inequality holds with exponent p if it holds with exponent for the truncated and centered random variables. The Bahr-Esseen inequality is also true if the truncated random variables are acceptable. We then apply the results to obtain weak and strong laws of large numbers and complete convergence.

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Exponential inequalities are obtained under general conditions. Then it is shown that an exponential inequality implies both Rosenthal's inequality and complete convergence of sums of random variables. The general results are applied to weakly orthant dependent sequences.

We study the eigenvalues of large perturbed matrices. We consider an Hermitian pattern matrix of rank . We blow up to get a large blockmatrix . Then we generate a random noise and add it to the blown up matrix to obtain the perturbed matrix = + . Our aim is to find the eigenvalues of . We obtain that under certain conditions has 'large' eigenvalues which are called structural eigenvalues. These structural eigenvalues of approximate the non-zero eigenvalues of . We study a graphical method to distinguish the structural and the non-structural eigenvalues. We obtain similar results for the singular values of non-symmetric matrices.