  In this paper, we show that the adaptive multidimensional increment ratio
estimator of the long range memory parameter defined in Bardet and Dola (2012)
satisfies a central limit theorem (CLT in the sequel) for a large
semiparametric class of Gaussian fractionally integrated processes with memory
parameter $d \in (-0.5,1.25)$. Since the asymptotic variance of this CLT can be
computed, tests of stationarity or nonstationarity distinguishing the
assumptions $d<0.5$ and $d \geq 0.5$ are constructed. These tests are also
consistent tests of unit root. Simulations done on a large benchmark of short
memory, long memory and non stationary processes show the accuracy of the tests
with respect to other usual stationarity or nonstationarity tests (LMC, V/S,
ADF and PP tests). Finally, the estimator and tests are applied to log-returns
of famous economic data and to their absolute value power laws.
