Journal of Multivariate Analysis
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In a previous paper, the behavior of the subdominant eigenvalue of matrices $B=(b_{i,j})\in \mathbb{R}^{n,n}$ whose entries are independent random variables with an expectation E(bi,j)=1/n and with a variance ${\rm Var}(b_{i,j}) \leq c_1/n^2$, for some constant $c_1\geq 0$, was investigated. For such matrices it was shown that for large n, the subdominant eigenvalues of B are, with great probability, in a small neighborhood of 0. Here we replace the assumption that the individual entries of B are independent random variables with the weaker assumption that the rows of B are independent n-dimensional random variables but which, within each row, satisfy that $|{\rm Cov} (b_{i,j},b_{i,k})|\leq c_2/n^3$ for some constant $c_2\geq 0$. We show that under these conditions the subdominant eigenvalues of B continue to tend in probability to 0 as $n\to \infty$. Our assumptions are satisfied, for example, in the case that $B\in \mathbb{R}^{n,n}$ is a stochastic matrix whose rows are chosen from a certain simplex lying in $\mathbb{R}^n$ according to the symmetric Dirichlet distribution satisfying further certain stipulation. The n-dimensional uniform distribution arises as a special case of this stipulation.