This question is part of the original question: Convergence/divergence of the operator norm for random sign matrices.
For each ${n}$, let ${A_n = (a_{ij,n})_{1 \leq i,j \leq n}}$ be an ${n \times n}$ random sign matrix (i.e. the entries ${a_{ij,n}}$ of ${A_n}$ are jointly independent in ${i,j}$ and take values in ${\{-1,+1\}}$ with a probability of ${1/2}$ each). If ${\|A_n\|_{op}}$ denotes the operator norm of ${A_n}$, one can show that
$\displaystyle {\|A_n\|_{op} / n^{1/2+\varepsilon}} \leq [\hbox{tr}(A_nA_n^*)^k]^{\frac{1}{2k}} / n^{1/2+\varepsilon}$
I was trying to show that the quantity on the LHS converges to zero almost surely by setting $k=2$ (so that the trace expression for the symmetric matrix $\hbox{tr}(A_nA_n^*)^2$ becomes the sum of the square of the entries of $A_nA_n^*$), and hopefully getting the quantity $\hbox{tr}(A_nA_n^*)^2$ bounded by $O(n^2)$. Yet it didn't seem to work, is there any better way to do it?