Tables of finite-mean nonsymmetric stable distributions as computed from their convergent and asymptotic series†

Abstract

Stable distributions take their interest from the fact that they are the only possible limiting distributions for sums of independent identicalloy distributed random variable. The Gaussian and Cauchy distributions are the two best-known members of the family. Though stable distributions have long been known and studied, little use has been made of the non-Gaussian ones, especially those that are nonsymmetric as well, in mathematical models. One reason for this is that closed form expressions exist for only a very few stable densities. Alothough probability theory tells us much about the general behavior of the stables, particulary their asymptotic properties, to find out their specific behavior on the inner ranges involves tedious computation. In this paper we attempt to alleviate some of the problems of using finite mean nonsymmetric stables, by computing density graphs and cumulative distribution tables for them using their convergent and asymptotic series. A rough estimate of the scale parameter is also obtained.