# Random matrix with non-identical variances

Hello,

Consider $A$ a $n \times n$ random matrix with centered Gaussian entries $A_{i,j}$ such that $$\mathbb{E}[A_{i,j}^2]=\sigma_j^2/n$$. The variances depend on the column only.

What do we know on the eigenvalues distribution ? In particular, if one assumes that $$\frac{1}{n}\sum_{j=1}^n \sigma_j^2 \to \bar{\sigma}^2$$ then is it true that :

$\rho(A):=max(|\lambda_k|) \to \bar{\sigma}^2$ when $n\to \infty$ ?

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Not really an answer, but you can check out en.wikipedia.org/wiki/Circular_law and the references listed there. To my knowledge, nothing other than the IID case is known. – Stanley Yao Xiao Feb 24 '12 at 19:28

I'm not sure this answers your question exactly but there are some known bounds. For example, Latala provides the following high probability bound under the conditions of finite fourth moments for the matrix entries $$\|A\|_2 \leq C \left( \max_i \sqrt{\sum_j \mathbb{E}A_{ij}^2}+ \max_j \sqrt{\sum_i \mathbb{E}A_{ij}^2} + \sqrt[4]{\sum_{i,j}\mathbb{E}A_{ij}^4} \right).$$