  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 $T_n$,
which allows us to construct an estimator $\hat{k}$ of a change-point. We then
compute the statistic $T_{n,1}$ based on the observations up to time $\hat{k}$
and the statistic $T_{n,2}$ based on the observations after time $\hat{k}$. The
statistic $M_n=\max[T_{n,1},T_{n,2}]$ 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.
