## Brazilian Journal of Probability and Statistics

### Change in the mean in the domain of attraction of the normal law via Darling–Erdős theorems

#### Abstract

This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ versus the at most one change in the mean alternative hypothesis. The approach taken is via a Darling–Erdős type self-normalized maximal deviation between sample means before and sample means after possible times of a change in the expected values of the observations of a random sample. Asymptotically, the thus formulated maximal deviations are shown to have a standard Gumbel distribution under the null assumption of no change in the mean. A first such result is proved under the condition that $EX^{2}\log\log(|X|+1)<\infty$, while in the case of a second one, $X$ is assumed to be in a specific class of the domain of attraction of the normal law, possibly with infinite variance.

#### Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 4 (2014), 538-560.

Dates
First available in Project Euclid: 30 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1406741879

Digital Object Identifier
doi:10.1214/13-BJPS223

Mathematical Reviews number (MathSciNet)
MR3263064

Zentralblatt MATH identifier
1295.53077

#### Citation

Csörgő, Miklós; Hu, Zhishui. Change in the mean in the domain of attraction of the normal law via Darling–Erdős theorems. Braz. J. Probab. Stat. 28 (2014), no. 4, 538--560. doi:10.1214/13-BJPS223. https://projecteuclid.org/euclid.bjps/1406741879

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