Séries de potência formais para as distribuições estáveis de Lévy: o caso simétrico

Abstract

A relevant problem in Statistical Physics and Mathematical Physics is to derive numerically precise expressions and exact analytical forms to calculate the distributions of Lévy α-stable Pα(x; β). In practice, these distributions are usually expressed in terms of the Fourier Integral of its characteristic function. In fact, known closed-form expressions are relatively scarce given the huge space of parameters: 0 < α ≤ 2 (L´evy index), −1 ≤ β ≤ 1 (asymmetry), σ > 0 (scale) and −∞ < µ < ∞ (offset). In the formal context, important exact results rely on special functions, such as the Meijer-G, Fox-H functions and finite sum of hypergeometric functions, with only a few exceptional cases expressed in terms of elementary functions (Gaussian and Cauchy distributions). From a more practical point of view, methods such as, e.g., series expansions allow an estimation of the Lévy distributions with high numerical precision, but most of the approaches are restricted to a small subset of the parameters and, although sophisticated, these algorithms are time-consuming. As an additional contribution to this problem, we propose new methods to describe the symmetric stable distributions, with parameters β = 0, µ = 0, σ = 1. We obtain a description through a closed analytical form, via formal power series making use of the Borel regularization sum procedure (for α = 2/M, M = 1, 2, 3... ). Furthermore we obtain an approximate expression (for 0 < α ≤ 2) by dividing the domain of the integration variable into sub-intervals (windows), performing proper series expansion inside each window, and then calculating the integrals term by term.