doctoralThesis
Séries de potência formais para as distribuições estáveis de Lévy: o caso simétrico
Fecha
2018-08-17Registro en:
COSTA NETO, José Crisanto da. Séries de potência formais para as distribuições estáveis de Lévy: o caso simétrico. 2018. 106f. Tese (Doutorado em Física) - Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Natal, 2018.
Autor
Costa Neto, José Crisanto da
Resumen
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.