On heavytail phenomena in some large deviations problems
Abstract
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 heavytail 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 lastpassage 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 $\alpha \in (0,1)$.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 arXiv:
 arXiv:1706.06184
 Bibcode:
 2017arXiv170606184A
 Keywords:

 Mathematics  Probability