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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 $p=2,3,4$.

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up vote 6 down vote accepted

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 (and the back references, going to Nagaev and earlier).

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Thank you very much for the reference – mohi Apr 25 '14 at 17:24

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