Concentration of sum of powers of normals

Question

Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables i.e.~$X\sim\mathcal{N}(0,1)$ and $Y\sim\mathcal{N}(0,1)$. Simply stated $Z$ is a complex Gaussian random variable with $\mathbb{E}[|Z|^2]=1$. I am interested in concentration of the form \begin{align} \mathbb{P}\left\{\Bigg|\frac{\sum_{k=1}^n|Z_k|^{2p}}{n}-p!\Bigg|>t\right\}<f(t,p,n). \end{align} (Please note that $E[|Z|^{2p}]=p!$). I would like a result as sharp as possible but "simple" that holds for small values of $t$. I would like the result to hold for arbitrary small values of $t$ e.g. for all $0< t<t_0$ for $t_0$ a fixed numerical constant. For example, I was wondering if one can prove the above for $f$ of the form below \begin{align} f(t,n,p)=ce^{-\gamma\sqrt[p]{nt}}, \end{align} where $c$ and $\gamma$ are fixed numerical constants. I should add that I don't need the result for all value of $p$ and would be happy with a result that holds for small values of $p$ such as $p=2,3,4$.

Answer 1

For $p>1$, the random variables you discuss do not possess exponential moments; You are in the regime of large deviations with stretched exponential tails. See for example the following recent paper by Gantert, Ramanan and Rembart http://arxiv.org/abs/1401.4577 (and the back references, going to Nagaev and earlier).