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Suppose that $g_1,\dots,g_m$ are i.i.d. $N(0,I_n)$ vectors. Let $$ X = \sum_{i\neq j} (\langle g_i,g_j \rangle^2 - n) $$ It is known that when $m\to \infty$, $n\to\infty$, $m/n\to 0$, $$ \frac{1}{2mn}X \to N(0,1) \quad \text{in distribution}. $$ My question is, how can we obtain a concentration bound for $X$, maybe something like $\Pr\{X > tmn\} \leq \exp(-ct^2)$? Is this possible?

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