Abstract:
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The topic is related to the Central Limit Theorem. In this paper, we study the self-normalized moderate deviations for centered independent random variables with finite third or higher moments. With these moment conditions, we obtain the exact self-normalized tail probabilities for all x in the range of [0, o(n^(1/4))]. This is an extension of the results in Jing, Shao and Wang (2003) where at most finite third moment is assumed. In particular, if the centered independent random variables have some third moment conditions, the self-normalized moderate deviation probabilities hold uniformly in a range of x which is related to the moments with order between 3 and 4. Furthermore, it is proved that the range [0, o(n^(1/4))] is optimal under these third moment conditions. At the end, we show the necessity of these third moment conditions in obtaining the self-normalized moderate deviation probabilities for x outside the range of [0, o(n^(1/6))].
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