Is there a good tail bound for $\operatorname{P}\!\Bigg[\bigg\vert\dfrac{\sum_{j=1}^n(\sum_{i=1}^n a_{i,j})^2}{n^2} -1\bigg\vert > \epsilon\Bigg]\,,$ where all $a_{i,j}$'s are iid, with $\operatorname{E}[a_{i,j}] = 0\,, \operatorname{E}[a_{i,j}^2] = 1\,, \operatorname{E}[|a_{i,j}|^3] = \rho\,.$

The tail bound can be without any relation to $\epsilon$ but needs to converge to 0 as $n\to\infty\,.$