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

TL;DR

This work sharpens finite-sample central limit bounds for i.i.d. sums with moment order $r=2+\delta$, $\delta\in[0,1]$, by replacing moment-based error terms with a weak distance to normality measured by Zolotarev–Senatov norms $\zeta_{m,g}$. It establishes a Senatov–Zolotarev type bound, $\zeta_1(\widetilde{P^{\ast n}}-\mathrm{N}) \le n^{-\delta/2} \xi_{\delta}( \zeta_1(\widetilde{P}-\mathrm{N}), \zeta_{2,\delta}(\widetilde{P}-\mathrm{N}) )$, and then derives the Lyapunov–Katz bound in the Kolmogorov metric, $\|\widetilde{P^{\ast n}}-\mathrm{N}\|_{\mathrm{K}} \le \frac{c}{n^{\delta/2}} ( \zeta_1 \vee \zeta_{2,\delta} )(\widetilde{P}-\mathrm{N})$, for $n\ge 2$, with a universal constant $c$ (e.g., $c=48$). The proofs combine a refined smoothing argument, a convolution inequality, and a careful analysis of the $\zeta_{m,g}$ norms, relying on a partial generalisation of Senatov–Zolotarev theory and clarifying historical attributions and gaps (including errata to Mattner2024). The results extend Mattner (2024) by treating the full range $\delta\in[0,1]$ and improving the dependence on normal closeness, with explicit constants and a groundwork of auxiliary lemmas for the $\zeta_{m,g}$ framework.

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.

Theorems & Definitions (18)

• Theorem 1.1: Lyapunov-Katz for i.i.d. summands Zolotarev-close to normal
• Theorem 1.2: A Senatov-Zolotarev type $\zeta_{1} \vee \zeta_{2,\delta}$ Theorem
• proof : Proof of Theorem \ref{['Thm--Z/S_zeta1_zeta2,delta']}
• proof : Proof of Theorem \ref{['Thm:Lyapunov-Katz_for_summands_Z-close_to_normal']}
• Lemma 3.1
• proof
• Lemma 3.2: A scaling inequality for $\zeta_{2,\delta}$
• proof
• Lemma 3.3
• proof