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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 $n\to\infty\,.$

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    $\begingroup$ Erm... What is the event in brackets? $\endgroup$ – fedja May 16 at 15:40
  • $\begingroup$ Is that meant to be $P[...-1]>0$? $\endgroup$ – Bullet51 May 17 at 2:08
  • $\begingroup$ Sorry for the mistake. I have updated. $\endgroup$ – 金之涵 May 17 at 2:13
  • $\begingroup$ It is unclear what you mean by poly(n) and polylogn; please use standard mathematical notation. It is also unclear if you mean an upper or lower bound. If you mean an upper bound that goes to $0$ as $n\to\infty$, for each $\epsilon>0$, then such a bound does not exist. $\endgroup$ – Iosif Pinelis 2 days ago
  • $\begingroup$ Why doesn't it exist? $\endgroup$ – 金之涵 2 days ago

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