The Annals of Probability

Asymptotic Normality for a General Statistic from a Stationary Sequence

Edward Carlstein

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Abstract

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 $t_n$.

Article information

Source
Ann. Probab., Volume 14, Number 4 (1986), 1371-1379.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992377

Digital Object Identifier
doi:10.1214/aop/1176992377

Mathematical Reviews number (MathSciNet)
MR866357

Zentralblatt MATH identifier
0609.62025

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60G10: Stationary processes

Keywords
Asymptotic normality general statistic $\alpha$-mixing sample fractiles stationary sequence

Citation

Carlstein, Edward. Asymptotic Normality for a General Statistic from a Stationary Sequence. Ann. Probab. 14 (1986), no. 4, 1371--1379. doi:10.1214/aop/1176992377. https://projecteuclid.org/euclid.aop/1176992377


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