Exponential Concentration Inequality for Higher-order moments of Gaussian Random Variables

Question

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| >t\Bigr) &\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| >t\Bigr) &\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| >t\Bigr) \leq 2 \exp( cnt)? \end{align} And more generally, for a centered random variable $Y$ satisfying $\mathbb{P}(|Y| > t) \leq 2 \exp(c t^{\alpha})$ for $\alpha > 0$, 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| >t\Bigr) \leq 2 \exp( cnt^\beta) \end{align} for some $\beta > 0$？

Answer 1

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.