The Annals of Statistics

On discriminating between long-range dependence and changes in mean

István Berkes, Lajos Horváth, Piotr Kokoszka, and Qi-Man Shao

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We develop a testing procedure for distinguishing between a long-range dependent time series and a weakly dependent time series with change-points in the mean. In the simplest case, under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point, and under the alternative it is long-range dependent. We compute the CUSUM statistic Tn, which allows us to construct an estimator of a change-point. We then compute the statistic Tn,1 based on the observations up to time and the statistic Tn,2 based on the observations after time . The statistic Mn=max [Tn,1,Tn,2] converges to a well-known distribution under the null, but diverges to infinity if the observations exhibit long-range dependence. The theory is illustrated by examples and an application to the returns of the Dow Jones index.

Article information

Ann. Statist. Volume 34, Number 3 (2006), 1140-1165.

First available in Project Euclid: 10 July 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G10: Hypothesis testing

Change-point in mean CUSUM long-range dependence variance of the mean


Berkes, István; Horváth, Lajos; Kokoszka, Piotr; Shao, Qi-Man. On discriminating between long-range dependence and changes in mean. Ann. Statist. 34 (2006), no. 3, 1140--1165. doi:10.1214/009053606000000254.

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