Infinitely Divisible Limit Processes for the Ewens Sampling Formula |
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Authors: | Babu G. J. Manstavičius E. |
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Affiliation: | (1) Department of Statistics, The Pennsylvania State University, 319 Thomas Building, University Park, PA 16802, USA;(2) Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-2600 Vilnius, Lithuania |
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Abstract: | The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a partial sum process weakly converges in a function space if and only if the corresponding process defined through sums of independent random variables weakly converges. As a consequence of this result, necessary and sufficient conditions for weak convergence to a stable process are established. A counterexample showing that these conditions are not necessary for the one-dimensional convergence is presented. Very few results on the necessity part are known in the literature. |
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Keywords: | random partition population genetics allelic partition permutation probabilistic number theory Skorokhod topology functional limit theorem stable process |
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