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Limiting Distributions for a Class Of Diminishing Urn Models
- Part of
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- Journal:
- Advances in Applied Probability / Volume 44 / Issue 1 / March 2012
- Published online by Cambridge University Press:
- 04 January 2016, pp. 87-116
- Print publication:
- March 2012
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- Article
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In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix
and c = pa with 
. We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary 
and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.
and 
. We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem 
and 