Necessary and Sufficient Conditions for $L^p$ Convergence of the Largest Eigenvalue of a Wigner Matrix

Question

I am looking for a reference of the proof of the above claim.

Basically it's the following claim:

Consider a Wigner matrix $X_N$ satisfying $r_k\le k^{Ck}$ for some constant $C$ and all positive integers $k$. Then, $\lambda_N^N$ converges to $2$ in $L^p$ norm.

where $r_k$ is defined as: $r_k := \max ( E|Z_{1,2}|^k,E|Y_1|^k )<\infty$; and $\{ Z_{i,j} \}_{1\le i <j}$ and $\{ Y_i \}_{i\ge 1}$ are real valued random variables iid with zero mean, s.t $E(Z_{1,2}^2)=1$.

Thanks!

Answer 1

I believe this follows from the standard estimates that show the convergence. For example, look at the proof in Anderson-Guionnet-Zeitouni's book, page 24. Indeed, from the last display there, you get that $P(\lambda_N^N>(2+\delta))\leq e^{-\delta N^c}$ for an appropriate $c$, which is more than enough to conclude that $(\lambda_N^N/2)^p$ is uniformly integrable.