Estrutura analítica complexa das distribuições estáveis de Lévy
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-stabl...
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| Formato: | doctoralThesis |
| Idioma: | pt_BR |
| Publicado em: |
Universidade Federal do Rio Grande do Norte
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| Endereço do item: | https://repositorio.ufrn.br/handle/123456789/31016 |
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| Resumo: | 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. |
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