Deciding between I(0) and I(1) via FLIL-based bounds

We construct properly scaled functions of R-p-valued partial sums of demeaned data and derive bounds via the functional law of the iterated logarithm for strong mixing processes. If we obtain a value below or equal to the bound we decide in favor of I(0); otherwise we decide in favor of I(1), This provides a consistent rule for classifying time series as being I(1) or I(0), The nice feature of the procedure lies in the almost sure nature of the bound, guaranteeing a lim sup-type result. We finally provide conditions for the strong consistency of estimators of the variance in the dependent and heterogeneous case.