Estrutura analítica complexa das distribuições estáveis de Lévy

Abstract

The almost ubiquitous Lévy α-stable distributions lack general closed-form expressions in terms of elementary functions-Gaussian and Cauchy cases being notable exceptions. To better understand this 80-year-old conundrum, we study the complex analytic continuation pα(z), z ∈ C , of the Lévy α-stable distribution family pα(x), x ∈ R, parametrized by 0 < α ≤ 2. We first extend known but intricate results, and give a new proof that pα(z) is holomorphic on the entire complex plane for 0 < α ≤ 2, whereas pα(z) is not even meromorphic on C for 0 < α < 1. Next, we unveil the complete complex analytic structure of pα(z) using domain coloring. Finally, motivated by these insights, we argue that possibly, there cannot be closed-form expressions in terms of elementary functions for pα(x), for general α.