Probability Theory and Related Fields

, Volume 120, Issue 2, pp 236-254

First online:

A non-uniform Berry–Esseen bound via Stein's method

  • Louis H.Y. ChenAffiliated withDepartment of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore. e-mail:
  • , Qi-Man ShaoAffiliated withDepartment of Mathematics, University of Oregon, Eugene, OR 97403, USA. e-mail:

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments.

Mathematics Subject Classification (2000): Primary 60F05; Secondary 60G50
Key words or phrases: Stein's method – Normal approximation – Non-uniform Berry–Esseen bound – Concentration inequality approach