Zero Biasing and Jack Measures
نویسندگان
چکیده
Zero biasing for the normal approximation of a random variable W using Stein’s method was introduced in Goldstein and Reinert [16]. One instance in which the zero bias method may be applied is for W for which a Stein pair W,W ′ may be constructed, that is, for W that may be coupled to a variable W ′ such that W,W ′ is exchangeable and satisfies E(W ′|W ) = (1− a)W for some a ∈ (0, 1]. After giving a brief review of these methods in Section 2, in Section 3 we provide a general result allowing one to apply zero biasing when the statistic W of interest is formed by certain growth processes and can be coupled in a Stein pair. Section 4 studies a certain statistic Wα under the Jackα measure on partitions. We defer precise definitions to Section 4, but for now mention that it is of interest to study statistical properties of the Jackα measure. The case α = 1 corresponds to the actively studied Plancherel measure of the symmetric group. The surveys [1], [5] and [25] and the seminal papers [2], [20] and [24] indicate how the Plancherel measure of the symmetric group is a discrete analogue of random matrix theory, and describe its importance in representation theory and geometry. Okounkov [25] notes that the study of the Jackα measure is an important open problem, about which relatively little is known. It is a discrete analogue of Dyson’s β ensembles from random matrix theory [3]. The particular statistic Wα under the Jack measure which we study is of interest for several reasons. When α = 1 it reduces to the character ratio of transpositions under the Plancherel
منابع مشابه
Zero Biasing and Growth Processes
The tools of zero biasing are adapted to yield a general result suitable for analyzing the behavior of certain growth processes. The main theorem is applied to prove central limit theorems, with explicit error terms in the L metric, for certain statistics of the Jack measure on partitions and for the number of balls drawn in a Pólya-Eggenberger urn process.
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عنوان ژورنال:
- Combinatorics, Probability & Computing
دوره 20 شماره
صفحات -
تاریخ انتشار 2011