Stein's method using approximate zero bias couplings with applications to combinatorial central limit theorems under the Ewens distribution
Abstract
We generalize the well-known zero bias distribution and the $\lambda$-Stein pair to an approximate zero bias distribution and an approximate $\lambda,R$-Stein pair, respectively. Berry Esseen type bounds to the normal, based on approximate zero bias couplings and approximate $\lambda,R$-Stein pairs, are obtained using Stein's method. The bounds are then applied to combinatorial central limit theorems where the random permutation has the Ewens $\mathcal{E}_\theta$ distribution with $\theta>0$ which can be specialized to the uniform distribution by letting $\theta=1$. The family of the Ewens distributions appears in the context of population genetics in biology.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2017
- DOI:
- 10.48550/arXiv.1707.03546
- arXiv:
- arXiv:1707.03546
- Bibcode:
- 2017arXiv170703546W
- Keywords:
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- Mathematics - Probability;
- 60F05;
- 60C05
- E-Print:
- 30 pages