arXiv 1409.53962014-09math.PRmath.CO

秩一方差型随机矩阵的谱矩

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

Victor M. Preciado, M. Amin Rahimian

摘要：

设$\mathbf{a}_{ij}$、$1\leq i\leq j\leq n$为独立随机变量，对于所有$i,j$，设$\mathbf{a}_{ji}=\mathbf{a}_{ij}$为独立随机变量。假设对于给定的正实数序列$Ψ=\{σ_{i} , i\in\mathbb{N}\}$，每个$\mathbf{a}_{ij}$是有界的，均值为零，方差由$σ_{i}σ_{j}$给出。因此，方差矩阵$V_n=\left( \mbox{Var}( \mathbf{a}_{ij}) \right)_{i,j=1}^n$对于所有$n$具有秩1。我们发现，经验谱分布的对称随机矩阵$\mathbf{A}_n(Ψ)$ $ =$ $ \left( \mathbf{a}_{ij}/\sqrt{n}\right)_{i,j=1}^n$收敛弱（概率为1），我们充分表征了一个确定性的极限谱分布，其极限谱矩序列$Ψ$提供封闭形式的表达式。此外，我们提出了一个层次的半定程序计算上，下界的期望谱范数的$\mathbf{A}_n$，有限$n$和极限$n\to\infty$。

英文摘要：

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 $σ_{i}σ_{j}$, for a given sequence of positive real numbers $Ψ=\{σ_{i} , i\in\mathbb{N}\}$. 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 $\mathbf{A}_n(Ψ)$ $ =$ $ \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 $Ψ$. 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 $n\to\infty$.

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