Abstract
Applying an inductive technique for Stein and zero bias couplings yields Berry-Esseen theorems for normal approximation for two new examples. The conditions of the main results do not require that the couplings be bounded. Our two applications, one to the Erdős-Rényi random graph with a fixed number of edges, and one to Jack measure on tableaux, demonstrate that the method can handle non-bounded variables with non-trivial global dependence, and can produce bounds in the Kolmogorov metric with the optimal rate.
Highlights
We present new Berry-Esseen theorems for sums Y of possibly dependent variables by combining both the Stein and zero bias couplings of Stein’s method with the inductive technique of Bolthausen (1984) originally developed for the combinatorial central limit theorem
We develop results that can be applied to the Stein couplings of Chen and Rollin (2010) and to the zero bias couplings of Goldstein and Reinert (1997), encompassing most of the known couplings that have appeared in the literature, including settings not typically framed in terms of couplings, such as local dependence
This work is a broad extension and continuation of Ghosh (2009), applying induction and the zero bias coupling for the combinatorial central limit theorem where the random permutations are involutions, and of Goldstein (2013) using the size bias coupling to study degree counts in the AMS 2000 subject classifications: Primary 60F05; secondary 05C07, 05C80, 05E10 Keywords: Kolmogorov distance, optimal rates, Erdos-Renyi random graph, Jack measure
Summary
We present new Berry-Esseen theorems for sums Y of possibly dependent variables by combining both the Stein and zero bias couplings of Stein’s method with the inductive technique of Bolthausen (1984) originally developed for the combinatorial central limit theorem. Before attempting to apply the methods presented in this article, it is advisable that a user first ‘test the waters’ by constructing a Stein or zero-bias coupling and proving a normal approximation for a smooth metric such as the Wasserstein distance; see Chen and Rollin (2010), or Goldstein (2007), respectively. Once this goal has been achieved, the sigma-algebra Fθ will typically arise naturally from the coupling construction, and one may proceed to identify a suitable variable V whose conditional distribution given Fθ is within the same class of distributions determined by Θ and close to that of Y. In occupancy problems, a Stein coupling or zero-bias coupling typically involves moving around a small number of balls among a small number of urns, and V will typically again represent an occupancy problem, but on fewer balls and fewer urns
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