<p>Statistical theory for the estimators of the location, scale and shape parameters of the approximating Generalized Pareto distribution in the Peaks-Over-Threshold approach to extreme value statistics typically requires a second-order extended regular variation condition on the tail quantile function of the distribution underlying the random variable of interest. Somewhat surprisingly, the existing sufficient criteria ensuring that this condition holds appear not to easily apply to many common light-tailed distributions, from the well-known Gaussian, log-normal and Gamma distributions to more specific examples arising in finance and reliability theory. We provide several sufficient criteria based on the computation of appropriate asymptotic expansions of a distribution function or of its quantile function when the underlying distribution has a Weibull-type tail. A list of examples to which our theory applies is given.</p>