On Heavy-Tail Phenomena in Some Large-Deviations Problems

We revisit the proof of the large-deviations principle of Wiener chaoses partially given by Borell 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 fnDouble-struck capital Rn -> X, where X is some metric space, under the n-fold probability measure nu alpha n, where nu alpha=Y alpha-1e-x alpha dx alpha is an element of (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, in works by Bordenave and Caputo on one hand, and the author on the other hand, of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large-deviations result to the last-passage time, which yields a large-deviations principle when the weights follow the law Z alpha-1e-x alpha 1x >= 0dx, with alpha is an element of (0, 1).