Exponential Concentration Inequality for Higher-order moments of Gaussian Random Variables Let $X_1,\ldots, X_n$ be $n$ i.i.d. copies of  Gaussian random variable $X \sim N(0, \sigma^2)$. It is known that 
\begin{align}
\mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n X_j \Bigl| &gt;t\Bigr) &amp;\leq 2 \exp( cnt^2)~~\text{and}\\
\mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n (X_j^2 - \mathbb{E}X_j^2) \Bigl| &gt;t\Bigr) &amp;\leq 2 \exp( cn\min \{t^2, t\}).
\end{align}
These two results follows from the concentration inequality of sub-Gaussian and sub-exponential random variables and the second one is a Bernstein-type inequality. I wonder if there are similar results for the higher moments of Gaussian random variables. Can one have 
\begin{align}
\mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n (X_j^4 - \mathbb{E}X_j^4) \Bigl| &gt;t\Bigr) \leq 2 \exp( cnt)?
\end{align}
And more generally, for a centered random variable $Y$ satisfying $\mathbb{P}(|Y| &gt; t) \leq 2 \exp(c t^{\alpha})$ for $\alpha &gt; 0$, can we get an exponential inequality for the concentration of $\sum_{i=1}^n Y_i$? That is, can we have 
\begin{align}
\mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n (Y_j - \mathbb{E}Y_j) \Bigl| &gt;t\Bigr) \leq 2 \exp( cnt^\beta)
\end{align}
for some $\beta &gt; 0$？
See Theorem 23 in Section 9.3 of Ryan O'Donnell's book Analysis of Boolean functions. Even though the theorem there is stated for $\pm 1$ variables, it holds for Gaussians as well (see Chapter 10 of the book for details on that).

For the more general statement, see Exercise 7 in these lecture notes of Terry Tao.