An efficient series approximation for the Lévy α-stable symmetric distribution

A relevant problem in the statistical and mathematical physics literature is to derive numerically accurate expressions to calculate Lévy α-stable distributions Pα(x;β). On the formal side, important exact results rely on special functions, such as Meijer-G, Fox-H and finite sums of hypergeometric functions, with only a few exceptional cases expressed in terms of elementary functions. Hence, from a more practical point of view, methods such as series expansions are in order, e.g., to allow for the estimation of the Lévy distribution with high numerical precision, even though most of the existing approaches are restricted to a subset of the distribution parameters and/or usually demand relatively time-consuming sophisticated algorithms. Here we present a rather simple truncated expansion for the case of symmetric Lévy distributions Pα(x) (β=0). This is achieved by dividing the full range of integration into windows, performing proper series expansion inside each, and then calculating the integrals term by term. The obtained representation is convergent for any 0<α≤2. Moreover, its accuracy is directly controlled by the number of terms in the truncated expression, being straightforward to implement numerically. As we show with different examples, for almost all allowable α's the calculations lead to Pα(x) with reasonable low absolute error for computationally inexpensive simulations.