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I was wondering what is the best tail bound for \begin{equation*} \mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ? \end{equation*} where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.

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What counts as "best"? The smallest tail bound is of course $$ (2\pi)^{-n/2} \int_{\{(x_1,\dotsc,x_n) : \sum x_i^4 > (1+t)3n\} } e^{-\sum x_i^2 / 2} dx_1 \dotsb dx_n. $$

Presumably you want something simpler. Using standard concentration inequalities, $$ \mathbb{P} \left\{ \|X\|_4 > \mathbb{E}\|X\|_4 + s \right\} \le e^{-s^2/2} $$ where $\|X\|_4^4 = \sum_{k=1}^n X_k^4$, and $\mathbb{E}\|X\|_4 \sim (3n)^{1/4}$. It would take some fiddling to get the best constants, but assuming you're interested in large $t$, you would get an upper bound like $C \exp[-c\sqrt{(nt)}]$.

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I need the results to hold for all t>0. Is there a result that applies in that setting? –  mohi Apr 21 at 20:55
    
@mohi: Sure, everything I wrote prior to the last sentence holds for all t. You could also supplement the final bound I wrote (which would apply for $t > C$ with a simple Markov-type bound for smaller $t$. –  Mark Meckes Apr 22 at 4:57

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