A question from Zeitouni's Introduction to Random Matrices

Question

I have a question regarding exercise 2.1.5 on page 19 in this book: http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf

I would like a reference or help on this exercise.

The exercise asks the following:

Consider symmetric random matrices $X_N$, with the zero mean independent random varibles $\{ X_N(i,j) \}_{1\le i \le j\le N}$ no longer assumed identically distributed nor all of variance $1/N$. Check that Theorem 2.1.1 still holds if one assumes that for all $\epsilon >0 $, $$\lim_{N \to \infty} \frac{\#\{ (i,j) : |1-NE(X_N(i,j)^2)|< \epsilon \}}{N^2}=1$$

and for all $k\ge 1$, there exists a finite $r_k$ independent of $N$ s.t: $$\sup_{1\le i \le j\le N} E|\sqrt{N}X_N(i,j)|^k \le r_k$$

where Theorem 2.1.1 appears on page 7, and it states the following: For a Wigner matrix, the empirical measure $L_N$ converges weakly, in probability, to the semicircle distribution.

So how to show that the theorem stays the same under this change, is there any reference for this?

Thanks. P.S I tried to ask my question in MSE, but so far only 11 viewed it and none replied, it seems more appropriate here.

Answer 1

I think the essence is the central limit theorem. If you compute the traces of powers of your random matrix, they will be the sum of many independent random variables and will be Gaussian distributed when $N$ is large. So you should be able to show that these traces are not too far from the traces of the simpler case in this limit, implying the same limiting density. This is generally known as the method of moments.

You could have a look at the book by Terence Tao, "Topics in Random Matrix Theory", available here:

https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf