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!



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


