<p>See Theorem 23 in <a href="http://www.contrib.andrew.cmu.edu/~ryanod/?p=1472" rel="nofollow">Section 9.3 of Ryan O'Donnell's book Analysis of Boolean functions</a>. 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).</p>

<p>For the more general statement, see Exercise 7 in <a href="https://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/" rel="nofollow">these lecture notes of Terry Tao</a>.</p>

CC BY-SA 3.0
2015-11-22T10:55:57.867Z
Sorry the previous statement of the question is not correct. Now I've made some corrections. And what can we say for more general random variables with some decaying tail behavior？
Steve
CC BY-SA 3.0
2015-11-22T13:38:00.000Z
Your more general question is addressed in lecture notes of Terry Tao.
Yuval Filmus
CC BY-SA 3.0
2015-11-20T21:24:57.317Z
2015-11-22T15:28:28.730Z
2015-11-22T15:28:28.730Z
Yuval Filmus
<p>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 $\beta &gt; 0$？</p>

<pr.probability><st.statistics>
Exponential Concentration Inequality for Higher-order moments of Gaussian Random Variables
Steve
# Exponential Concentration Inequality for Higher-order moments of Gaussian Random Variables

(Asked by: username_0 on 2015-11-20)

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 $\beta > 0$？

**Tags:** pr.probability, st.statistics

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## 1 answers

### Answer by username_1 on 2015-11-20. Score: 7

See Theorem 23 in [Section 9.3 of Ryan O'Donnell's book Analysis of Boolean functions](http://www.contrib.andrew.cmu.edu/~ryanod/?p=1472). 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](https://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/).

> **Comments:**
> **username_0**: Sorry the previous statement of the question is not correct. Now I've made some corrections. And what can we say for more general random variables with some decaying tail behavior？  
> **username_1**: Your more general question is addressed in lecture notes of Terry Tao.  

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