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Strong large deviations for arbitrary sequences of random variables

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  • Cyrille Joutard

Abstract

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. Copyright The Institute of Statistical Mathematics, Tokyo 2013

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  • Cyrille Joutard, 2013. "Strong large deviations for arbitrary sequences of random variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 49-67, February.
    • Handle: RePEc:spr:aistmt:v:65:y:2013:i:1:p:49-67
    DOI: 10.1007/s10463-012-0361-1
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    References listed on IDEAS

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    1. Djamal Louani, 1998. "Large Deviations Limit Theorems for the Kernel Density Estimator," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 243-253, March.
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    Cited by:

    1. Joutard, Cyrille, 2014. "Asymptotic approximation for the probability density function of an arbitrary sequence of random variables," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 100-107.
    2. Cyrille Joutard, 2017. "Multidimensional strong large deviation results," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(6), pages 663-683, November.

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