### 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.

Original language | English (US) |
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Pages (from-to) | 232-242 |

Number of pages | 11 |

Journal | Lithuanian Mathematical Journal |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Lithuanian Mathematical Journal*,

*42*(3), 232-242. https://doi.org/10.1023/A:1020265607917

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*Lithuanian Mathematical Journal*, vol. 42, no. 3, pp. 232-242. https://doi.org/10.1023/A:1020265607917

**Infinitely divisible limit processes for the Ewens sampling formula.** / Babu, G. Jogesh; Manstavičius, E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Infinitely divisible limit processes for the Ewens sampling formula

AU - Babu, G. Jogesh

AU - Manstavičius, E.

PY - 2002/1/1

Y1 - 2002/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=53949123268&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53949123268&partnerID=8YFLogxK

U2 - 10.1023/A:1020265607917

DO - 10.1023/A:1020265607917

M3 - Article

AN - SCOPUS:53949123268

VL - 42

SP - 232

EP - 242

JO - Lithuanian Mathematical Journal

JF - Lithuanian Mathematical Journal

SN - 0363-1672

IS - 3

ER -