Heavy tailed distributions which allow for values far from the mean to occur with considerable probability are of increasing importance in various applications as the arsenal of analytical and numerical tools grows. Examples of interest are the Stable and more generally the Pareto distributions for which moments of sufficiently large order diverge. In fact, the asymptotic powerlaws of the distribution function at infinity and zero are directly related to the existence of positive and negative order moments, respectively. In practice, however, when dealing with finite size data sets of an unknown distribution, standard empirical estimators of moments will typically fail to reflect the theoretical divergence of moments and provide finite estimates for all order moments. The main contribution of this paper is an empirical wavelet-based estimator for the characteristic exponents of a random variable, which bound the interval of all orders r with finite moment. This objective is achieved by deriving a theoretical equivalence between finiteness of moments and the local Hölder regularity of the character- istic function at the origin and by deriving a wavelet based estimation scheme which is particularly suited to characteristic functions.