  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 $\alpha \in (0,1)$.
