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April, 1978 Some $L_p$ Versions for the Central Limit Theorem
Makoto Maejima
Ann. Probab. 6(2): 341-344 (April, 1978). DOI: 10.1214/aop/1176995580

Abstract

Let $\bar{F}_n(x)$ denote the distribution of the normalized partial sum of independent, identically distributed random variables with finite second moment, and write $\Delta_n(x) = |\bar{F}_n(x) - \Phi(x)|$, where $\Phi(x)$ is the standard normal distribution. In this paper, the necessary and sufficient conditions for the validity of $\|(1 + |x|)^{2 - 1/p}\Delta_n(x)\|_p = O(n^{-\delta/2})$ and of $\sum n^{-1 + \delta/2}\|(1 + |x|)^{2 - 1/p}\Delta_n(x)\|_p < \infty, 0 < \delta < 1, 1 \leqq p \leqq \infty$, are given. Furthermore, in the case where the underlying random variables $\{X_k\}$ are independent but not necessarily identically distributed, it is shown that $E|X_k|^{2 + \delta} < \infty$ implies $\|(1 + |x|)^{2 + \delta - 1/p}\Delta_n(x)\|_p \leqq Cs_n^{-(2 + \delta)} \sum^n_{k = 1} E|X_k|^{2 + \delta}, 0 < \delta < 1, 1 \leqq p \leqq \infty$.

Citation

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Makoto Maejima. "Some $L_p$ Versions for the Central Limit Theorem." Ann. Probab. 6 (2) 341 - 344, April, 1978. https://doi.org/10.1214/aop/1176995580

Information

Published: April, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0389.60014
MathSciNet: MR501281
Digital Object Identifier: 10.1214/aop/1176995580

Subjects:
Primary: 60F05
Secondary: 60F10

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 2 • April, 1978
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