Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 3, 312--327

BROWNIAN MOTION FOR RANDOM PERMUTATIONS

By

G.J. BABU, The Pennsylvania State University, University Park

and

E. MANSTAVICIUS, Vilnius University, Vilnius

SUMMARY. A family of measures, on the set of partitions of an integer, known as the Ewens sampling formula arises in population genetics. Mixtures of these measures also have applications in Bayesian statistics. Using methods from probabilistic number theory, a functional limit theorem in C [0,1] is established for a partial sum process based on these measures. The results can be used to develop statistical methods to test the validity of certain genetic models. It is interesting to note that a Lindeberg type condition is necessary for the dependent process to converge to the Brownian Motion, while it is not the case for the convergence of the one dimensional distributions. An example to illustrate this phenomenon is constructed.

AMS (1991) subject classification. Primary 60F17; secondary 60C05, 11K65.

Key words and phrases. Ewens sampling formula, random partitions, functional limit theorem, additive functions, multiplicative functions.

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