Strong large deviations for arbitrary sequences of random variables

We establish strong large deviation results for an arbitrary sequence of random variables under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic expansions for the tail probabilities of the same kind as those obtained by Bahadur and Rao (Ann. Math....

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Bibliographic Details
Published in:Annals of the Institute of Statistical Mathematics 2013-02, Vol.65 (1), p.49-67
Main Author: Joutard, Cyrille
Format: Article
Language:English
Subjects:
Density
Deviation
Economics
Finance
Insurance
Management
Mathematical analysis
Mathematics
Mathematics and Statistics
Probability
Random variables
Sample variance
Samples
Sampling
Standard deviation
Statistical analysis
Statistical methods
Statistics
Statistics for Business
Statistics Theory
Studies
Theorems
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Summary:We establish strong large deviation results for an arbitrary sequence of random variables under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic expansions for the tail probabilities of the same kind as those obtained by Bahadur and Rao (Ann. Math. Stat. 31:1015–1027, 1960 ) for the sample mean. We consider both the case where the random variables are absolutely continuous and the case where they are lattice-valued. Our proofs make use of arguments of Chaganty and Sethuraman (Ann. Probab. 21:1671–1690, 1993 ) who also obtained strong large deviation results and local limit theorems. We illustrate our results with the kernel density estimator, the sample variance, the Wilcoxon signed-rank statistic and the Kendall tau statistic.
ISSN:0020-3157
1572-9052
DOI:10.1007/s10463-012-0361-1