## Brazilian Journal of Probability and Statistics

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

### 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

**Keywords**

Change in the mean domain of attraction of the normal law Darling–Erdős theorems Gumbel distribution weighted metrics Brownian bridge

#### 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