We intend to start with a brief introduction to the classical Berry-Esseen theorem for sums of i.i.d. random variables. This is followed by suggesting in which form to teach, to any university students or even to sufficiently advanced high school pupils, the central limit theorem for the most classical discrete laws.

The main and more original part of the talk then presents an improvement of the Berry-Esseen theorem by taking a closeness to normality of the summands into account. We here improve results of Ulyanov (1976) and Senatov (1998), each previously optimal, in a line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our result works well, for example, for summands with a discretised or truncated normal law.

The talk is based on our long 2024 paper in Theory of Probability and Mathematical Statistics, on a further one with Lena Jonas to appear there in 2025, and on our 2025 arXiv preprint on teachable normal approximations.

Personal website of Lutz Mattner