Limit theorems for triangular urn schemes

We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depen...

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Bibliographic Details
Published inProbability theory and related fields Vol. 134; no. 3; pp. 417 - 452
Main Author JANSON, Svante
Format Journal Article
LanguageEnglish
Published Heidelberg Springer 01.03.2006
Berlin Springer Nature B.V
New York, NY
Subjects
Asymptotic properties
Distribution theory
Exact sciences and technology
Limit theorems
Mathematics
Probability
Probability and statistics
Probability theory and stochastic processes
Probability theory on algebraic and topological structures
Random variables
Sciences and techniques of general use
Statistics
Stochastic processes
Theorems
Triangular matrix
Urn scheme
Limit distribution
Probability theory
Distribution function
Asymptotic distribution
Stable law
Limit theorem
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Summary:We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depending on the ratio of the diagonal elements in the replacement matrix; the limit laws include normal, stable and Mittag-Leffler distributions as well as some less familiar ones. The results are in some cases similar to, but in other cases strikingly different from, the results for irreducible replacement matrices. [PUBLICATION ABSTRACT]
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-005-0442-7