TY - JOUR
T1 - A Functional Equation Whose Unknown is P([0,1]) Valued
AU - Aletti, Giacomo
AU - May, Caterina
AU - Secchi, Piercesare
PY - 2012/11
Y1 - 2012/11
N2 - 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.
AB - 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.
KW - Functional equation in unknown distribution functions
KW - Generalized Pólya urn
KW - Reinforced urn process
UR - http://www.scopus.com/inward/record.url?scp=84869092482&partnerID=8YFLogxK
U2 - 10.1007/s10959-011-0399-7
DO - 10.1007/s10959-011-0399-7
M3 - Article
SN - 0894-9840
VL - 25
SP - 1207
EP - 1232
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
IS - 4
ER -