In this paper, we revisit the proof of the large deviations principle of Wiener chaoses partially given by Borel, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Y_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the Wigner matrices without Gaussian tails, of the empirical spectral measure by Bordenave and Caputo, the largest eigenvalue and traces of polynomials by the author. We also apply our large deviations result to the last-passage time, which yields a large deviations principle when the weights have the density $Z_{\alpha}^{-1} e^{-x^{\alpha}}$ with respect to Lebesgue measure on $\mathbb{R}_+$, with $\alpha \in (0,1)$.

中文翻译：

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一些大偏差问题中的重尾现象

在本文中，我们重新审视了博雷尔部分给出的维纳混沌大偏差原理的证明，然后是勒杜的完整形式。我们表明，在大偏差中观察到的一些重尾现象可以用与维纳混沌相同的机制来解释，这意味着偏差在某种意义上是由平移产生的。更准确地说，我们证明了某一类泛函 $f_n 的一般大偏差原理： \mathbb{R}^n \to \mathcal{X}$，其中 $\mathcal{X}$ 是一些度量空间，在$n$-fold 概率测度 $\nu_{\alpha}^n$，其中 $\nu_{\alpha} =Y_{\alpha}^{-1}e^{-|x|^{\alpha}} dx$, $\alpha \in (0,2]$, 其中大偏差是由于平移。我们检索，作为一个应用程序，已知的无高斯尾的 Wigner 矩阵的大偏差原理，Bordenave 和 Caputo 的经验谱测量的最大特征值和作者的多项式迹。我们还将我们的大偏差结果应用于最后一段时间，当权重具有密度 $Z_{\alpha}^{-1} e^{-x^{\alpha}}$ 时，这会产生大偏差原理尊重 Lebesgue 对 $\mathbb{R}_+$ 的度量，其中 $\alpha \in (0,1)$。