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On necessary and sufficient conditions for the self-normalized central limit theorem

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Abstract

Let X1, X2, … be a sequence of independent random variables and Sn = Σ n i=1 Xi and V 2 n = Σ n i=1 X 2 i . When the elements of the sequence are i.i.d., it is known that the self-normalized sum Sn=Vn converges to a standard normal distribution if and only if max1⩽i⩽n|Xi|/Vn→0 in probability and the mean of X1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1⩽i⩽n|Xi|/Vn→0 in probability, then these sufficient conditions are necessary.

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Acknowledgements

This work was supported by Hong Kong Research Grants Council General Research Fund (Grant Nos. 14302515 and 14304917).

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Authors and Affiliations

  1. Department of Statistics, The Chinese University of Hong Kong, Hong Kong, China

    Qiman Shao

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  1. Qiman Shao
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Correspondence to Qiman Shao.

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In Honor of Professor Chuanrong Lu on His 85th Birthday

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Shao, Q. On necessary and sufficient conditions for the self-normalized central limit theorem. Sci. China Math. 61, 1741–1748 (2018). https://doi.org/10.1007/s11425-018-9368-3

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  • DOI: https://doi.org/10.1007/s11425-018-9368-3

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