Please use this identifier to cite or link to this item: http://cimat.repositorioinstitucional.mx/jspui/handle/1008/600
On the Non-Classical Infinite Divisibility of Power Semicircle Distribution
VICTOR MANUEL PEREZ ABREU CARRION
Acceso Abierto
Atribución-NoComercial
Probabilidad
Algebras - Operadores
Abstract. The family of power semicircle distributions de fi ned as normal- ized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian dis- tribution. A review of some results including a genesis and the so-called Poincaré’s theorem is presented. The moments of these distributions are re- lated to the super Catalan numbers and their Cauchy transforms in terms of hypergeometric functions are derived. Some members of this class of dis- tributions play the role of the Gaussian distribution with respect to additive convolutions in non-commutative probability, such as the free, the monotone, the anti-monotone and the Boolean convolutions. The in fi nite divisibility of other power semicircle distributions with respect to these convolutions is stud- ied using simple kurtosis arguments. A connection between kurtosis and the free divisibility indicator is found. It is shown that for the classical Gaussian distribution the free divisibility indicator is strictly less than 2.
Centro de Investigación en Matemáticas AC
17-03-2010
Reporte
Inglés
Investigadores
TEORÍA ANALÍTICA DE LA PROBABILIDAD
Versión publicada
publishedVersion - Versión publicada
Appears in Collections:Reportes Técnicos - Probabilidad y Estadística

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