Spectral Moments of Random Matrices with a Rank-One Pattern of Variances

Let $\mathbf{a}_{ij}$, $1\leq i\leq j\leq n$, be independent random variables and $\mathbf{a}_{ji}=\mathbf{a}_{ij}$, for all $i,j$. Suppose that every $\mathbf{a}_{ij}$ is bounded, has zero mean, and its variance is given by $\sigma_{i}\sigma_{j}$, for a given sequence of positive real numbers $\Psi=\{\sigma_{i} , i\in\mathbb{N}\}$. Hence, the matrix of variances $V_n=\left( \mbox{Var}( \mathbf{a}_{ij}) \right)_{i,j=1}^n$ has rank one for all $n$. We show that the empirical spectral distribution of the symmetric random matrix $\mathbf{A}_n(\Psi)$ $ =$ $ \left( \mathbf{a}_{ij}/\sqrt{n}\right)_{i,j=1}^n$ converges weakly (and with probability one) to a deterministic limiting spectral distribution which we fully characterize by providing closed-form expressions for its limiting spectral moments in terms of the sequence $\Psi$. Furthermore, we propose a hierarchy of semidefinite programs to compute upper and lower bound on the expected spectral norm of $\mathbf{A}_n$, for both finite $n$ and the limit $n\to\infty$.