Further refinement of self-normalized Cramér-type moderate deviations^∗

¹ Department of Mathematics, The University of Mississippi, University, MS 38677, Oxford, USA .
sang@olemiss.edu
² Division of Arts and Sciences, Mississippi State University at Meridian, Meridian, MS 39307, Oxford, USA
lg481@msstate.edu

Received: 29 January 2015
Revised: 6 September 2016
Accepted: 12 May 2017

Abstract

In this paper, we study the self-normalized Cramér-type moderate deviations for centered independent random variables X₁,X₂,... with 0 <E | X_i | ³< ∞. The main results refine Theorems 1.1 and 1.2 of Wang [Q. Wang, J. Theoret. Probab. 24 (2011) 307–329], the Berry−Esseen bound (2.11) and Corollaries 2.2 and 2.3 of Jing, et al. [B.Y. Jing, Q.M. Shao and Q. Wang, Ann. Probab. 31 (2003) 2167–2215] under stronger moment conditions.