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Limit theorems for triangular urn schemes

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Abstract.

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.

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Correspondence to Svante Janson.

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Janson, S. Limit theorems for triangular urn schemes. Probab. Theory Relat. Fields 134, 417–452 (2006). https://doi.org/10.1007/s00440-005-0442-7

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Keywords

  • Black Ball
  • Characteristic Function
  • Limit Theorem
  • Asymptotic Normality
  • Probability Generate Function