Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

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 $C \exp[-c\sqrt{(nt)}]$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.