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. $\mathcal{N}(0,1)$.



concentration of sums of fourth moment of normals
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 $C \exp[-c\sqrt{(nt)}]$.


