Probability Theory and Related Fields

, Volume 120, Issue 2, pp 236–254

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

  • Louis H.Y. Chen
  • Qi-Man Shao

DOI: 10.1007/PL00008782

Cite this article as:
Chen, L. & Shao, QM. Probab Theory Relat Fields (2001) 120: 236. doi:10.1007/PL00008782


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 

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Louis H.Y. Chen
    • 1
  • Qi-Man Shao
    • 2
  1. 1.Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore. e-mail:
  2. 2.Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. e-mail: shao@math.uoregon.eduUS

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