A sharper Lyapunov-Katz central limit error bound for i.i.d. summands Zolotarev-close to normal

Lena Jonas; Lutz Mattner

doi:10.1090/tpms/1245

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arXiv · 2503.06653

A sharper Lyapunov-Katz central limit error bound for i.i.d. summands Zolotarev-close to normal

Lena Jonas·Lutz Mattner

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Abstract

We prove a central limit error bound for convolution powers of laws with finite moments of order $r \in \mathopen]2,3\mathclose]$, taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of $r=3$ of the sharpening of the Berry (1941) - Esseen (1942) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov's (1901) theorem. Our proof uses a partial generalisation of the theorem of Senatov and Zolotarev (1986) used for the earlier special case. A result more general than our main one could be obtained by using instead another theorem of Senatov (1980), but unfortunately an auxiliary inequality used in the latter's proof is wrong.

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Lena Jonas, Lutz Mattner. 2025-03-09. A sharper Lyapunov-Katz central limit error bound for i.i.d. summands Zolotarev-close to normal. https://doi.org/10.1090/tpms%2F1245

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