On Combinatorial Central Limit Theorems with Different Underlying Permutations Via Approximate Zero Biasing

Wasamon Jantai () and Nathakhun Wiroonsri ()
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Wasamon Jantai: Chulalongkorn University
Nathakhun Wiroonsri: King Mongkut’s University of Technology Thonburi

Sankhya A: The Indian Journal of Statistics, 2025, vol. 87, issue 2, No 1, 301 pages

Abstract: Abstract We provide new bounds between a combinatorial statistic of the form $$\varvec{Y =\sum _{i=1}^n X_{i,\pi (i)}}$$ Y = ∑ i = 1 n X i , π ( i ) and the standard normal distribution, where $$\varvec{\{X_{i,j} \}_{i,j=1}^n}$$ { X i , j } i , j = 1 n are independent real valued random variables, and $$\varvec{\pi \in \mathcal {S}_n}$$ π ∈ S n , which is independent of $$\varvec{X_{i,j}}$$ X i , j and follows either the uniform or Ewens distribution. The family of the Ewens distributions appears in the context of population genetics in biology. The bounds are based on both $$\varvec{L^1}$$ L 1 and $$\varvec{L}^{\varvec{\infty }}$$ L ∞ distances under different assumptions. As our method, we apply the approximate zero bias approach via Stein’s method to obtain the bounds.

Keywords: Stein’s method; Permutations; Berry Esseen bounds; Wasserstein distance; Ewens distribution; Approximate zero bias coupling; Primary 60F05; Secondary 60C05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13171-025-00408-7

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