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October, 1986 Asymptotic Normality for a General Statistic from a Stationary Sequence
Edward Carlstein
Ann. Probab. 14(4): 1371-1379 (October, 1986). DOI: 10.1214/aop/1176992377


Let $\{Z_i: -\infty < i < + \infty\}$ be a strictly stationary $\alpha$-mixing sequence. Without specifying the dependence model giving rise to $\{Z_i\}$, and without specifying the marginal distribution of $Z_i$, we address the question of asymptotic normality for a general statistic $t_n(Z_1,\ldots, Z_n)$. The main theoretical result is a set of necessary and sufficient conditions for joint asymptotic normality of $t_n$ and a subseries value $t_m (m \leq n)$. Our theorems on asymptotic normality are the natural analogs to earlier results that deal with general statistics from iid sequences, and to other results that apply to the sample mean from dependent sequences. Asymptotic normality of the sample mean and of the sample fractiles follows as a special case of our general statistic $t_n$.


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Edward Carlstein. "Asymptotic Normality for a General Statistic from a Stationary Sequence." Ann. Probab. 14 (4) 1371 - 1379, October, 1986.


Published: October, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0609.62025
MathSciNet: MR866357
Digital Object Identifier: 10.1214/aop/1176992377

Primary: 62E20
Secondary: 60G10

Keywords: $\alpha$-mixing , asymptotic normality , general statistic , sample fractiles , stationary sequence

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 4 • October, 1986
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