  The one parameter family of Jack(alpha) measures on partitions is an
important discrete analog of Dyson's beta ensembles of random matrix theory.
Except for special values of alpha=1/2,1,2 which have group theoretic
interpretations, the Jack(alpha) measure has been difficult if not intractable
to analyze. This paper proves a central limit theorem (with an error term) for
Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we
recover a known central limit theorem on the distribution of character ratios
of random representations of the symmetric group on transpositions. The case
alpha=2 gives a new central limit theorem for random spherical functions of a
Gelfand pair. The proof uses Stein's method and has interesting ingredients: an
intruiging construction of an exchangeable pair, properties of Jack
polynomials, and work of Hanlon relating Jack polynomials to the Metropolis
algorithm.
