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.