Evaluating moments of length of Pitman partition

Abstract

Introduced in a seminal paper by Jim Pitman in 1995, the Pitman sampling formula has been intensively studied as a distribution of random partitions. One of the objects of interest is the length K(= K n,θ,α ) of a random partition that follows the Pitman sampling formula, where n ∈ N, α ∈ (0, ∞) and θ > −α are parameters. This paper presents asymptotic evaluations of E[K r ] (r = 1, 2, . . .) under two asymptotic regimes. In particular, the goals of this study are to provide a finer approximate evaluation of E[K r ] as n → ∞ than has previously been developed and to provide an approximate evaluation of E[K r ] as the parameters n and θ simultaneously tend to infinity with θ/n → 0. The results presented in this paper will provide a more accurate understanding of the asymptotic behavior of K.