Abstract
Let {X, X k : k ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution F. In this paper, the authors establish some results on the local precise large and moderate deviation probabilities for partial sums \({S_n} = \sum\limits_{i = 1}^n {{X_i}} \), in a unified form in which x may be a random variable of an arbitrary type, which state that under some suitable conditions, for some constants T > 0, a and τ > 1/2 and for every fixed γ > 0, the relation
holds uniformly for all x ≥ γn τ as n→∞, that is,
. The authors also discuss the case where X has an infinite mean.
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This work was supported by the National Natural Science Foundation of China (No. 11401415).
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Cheng, F., Li, M. Local precise large and moderate deviations for sums of independent random variables. Chin. Ann. Math. Ser. B 37, 753–766 (2016). https://doi.org/10.1007/s11401-016-1002-4
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DOI: https://doi.org/10.1007/s11401-016-1002-4
Keywords
- Local precise moderate deviation
- Local precise large deviation
- Intermediate regularly varying function
- O-regularly varying function