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

We prove a central limit error bound for convolution powers of laws with finite moments of order r ∈ ] 2 , 3 ] r ∈ ]2,3] , taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of r = 3 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.