# Is there a 1/poly(n) or 1/polylogn upper-bound for this tail bound?

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 $$n\to\infty\,.$$

New contributor

• Erm... What is the event in brackets? – fedja May 16 at 15:40
• Is that meant to be $P[...-1]>0$? – Bullet51 May 17 at 2:08
• Sorry for the mistake. I have updated. – 金之涵 May 17 at 2:13
• 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. – Iosif Pinelis 2 days ago
• Why doesn't it exist? – 金之涵 2 days ago