  We tackle the stationarity issue of an autoregressive path with a polynomial
trend, and we generalize some aspects of the LMC test, the testing procedure of
Leybourne and McCabe. First, we show that it is possible to get the asymptotic
distribution of the test statistic under the null hypothesis of
trend-stationarity as well as under the alternative of nonstationarity, for any
polynomial trend of order $r$. Then, we explain the reason why the LMC test,
and by extension the KPSS test, does not reject the null hypothesis of
trend-stationarity, mistakenly, when the random walk is generated by a unit
root located at $-1$. We also observe it on simulated data and we correct the
procedure. Finally, we describe some useful stochastic processes that appear in
our limiting distributions.
