  The danger of confusing long-range dependence with non-stationarity has been
pointed out by many authors. Finding an answer to this difficult question is of
importance to model time-series showing trend-like behavior, such as river
run-off in hydrology, historical temperatures in the study of climates changes,
or packet counts in network traffic engineering. The main goal of this paper is
to develop a test procedure to detect the presence of non-stationarity for a
class of processes whose $K$-th order difference is stationary. Contrary to
most of the proposed methods, the test procedure has the same distribution for
short-range and long-range dependence covariance stationary processes, which
means that this test is able to detect the presence of non-stationarity for
processes showing long-range dependence or which are unit root. The proposed
test is formulated in the wavelet domain, where a change in the generalized
spectral density results in a change in the variance of wavelet coefficients at
one or several scales. Such tests have been already proposed in
\cite{whitcher:2001}, but these authors do not have taken into account the
dependence of the wavelet coefficients within scales and between scales.
Therefore, the asymptotic distribution of the test they have proposed was
erroneous; as a consequence, the level of the test under the null hypothesis of
stationarity was wrong. In this contribution, we introduce two test procedures,
both using an estimator of the variance of the scalogram at one or several
scales. The asymptotic distribution of the test under the null is rigorously
justified. The pointwise consistency of the test in the presence of a single
jump in the general spectral density is also be presented. A limited
Monte-Carlo experiment is performed to illustrate our findings.
