Abstract
Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\). Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \). Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \), \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\). Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \). We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \), there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \), \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\). In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \), there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \), \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\).
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Acknowledgments
The authors are grateful to the Referees for their constructive, perceptive, and substantial comments and suggestions which enabled the authors to greatly improve the paper. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
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Li, D., Klesov, O. & Stoica, G. On the central limit theorem along subsequences of sums of i.i.d. random variables. Stat Papers 55, 1035–1045 (2014). https://doi.org/10.1007/s00362-013-0551-9
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DOI: https://doi.org/10.1007/s00362-013-0551-9