Journal of Theoretical Probability

, Volume 18, Issue 4, pp 891–909 | Cite as

On the Self-Normalized Cramér-type Large Deviation


For the self-normalized sum, \(S_n/V_n\), it is shown that \(P(S_n/V_n\ge x)/(1-\Phi(x))\) converges to 1, uniformly in a region, under the optimal assumption that the sampled distribution is in the domain of attraction of the normal law. Bounds for this convergence are given and their applications to exponential non-uniform Berry–Esseen bound are also discussed.

Key words

Cramér large deviation moderate deviation non-uniform Berry–Esseen bound domain of attraction of the normal law self-normalized sum 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of SydneySydneyAustralia

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