  We propose a procedure to decide between the null hypothesis of (strict)
stationarity and the alternative of non-stationarity, in the context of a
Random Coefficient AutoRegression (RCAR). The procedure is based on randomising
a diagnostic which diverges to positive infinity under the null, and drifts to
zero under the alternative. Thence, we propose a randomised test which can be
used directly and - building on it - a decision rule to discern between the
null and the alternative. The procedure can be applied under very general
circumstances: albeit developed for an RCAR model, it can be used in the case
of a standard AR(1) model, without requiring any modifications or prior
knowledge. Also, the test works (again with no modification or prior knowledge
being required) in the presence of infinite variance, and in general requires
minimal assumptions on the existence of moments.
