We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.

A Functional Equation Whose Unknown is P([0,1]) Valued / G. Aletti, C. May, P. Secchi. - In: JOURNAL OF THEORETICAL PROBABILITY. - ISSN 0894-9840. - 25:4(2012), pp. 1207-1232. [10.1007/s10959-011-0399-7]

A Functional Equation Whose Unknown is P([0,1]) Valued

G. Aletti
Primo
;
2012

Abstract

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.
Functional equation in unknown distribution functions; Generalized Pólya urn; Reinforced urn process
Settore MAT/06 - Probabilita' e Statistica Matematica
Settore SECS-S/01 - Statistica
http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10959-011-0399-7
http://hdl.handle.net/2434/62799
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/165655
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