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 k̂ of a change-point. We then compute the statistic Tn,1 based on the observations up to time k̂ and the statistic Tn,2 based on the observations after time k̂. 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.
"On discriminating between long-range dependence and changes in mean." Ann. Statist. 34 (3) 1140 - 1165, June 2006. https://doi.org/10.1214/009053606000000254