Testing the equality of the laws of two strictly stationary processes

In this paper we consider the problem of comparison two strictly stationary processes. The novelty our approach is that all their d-dimensional joint distributions, for $$d\geqslant 1$$ . Our procedure consists in expanding densities a multivariate orthogonal basis and comparing k first coefficients. dimension d to number coefficients compare view performing test can growth with sample size are automatically selected by two-step data-driven procedure. method works possibly paired, short or long range dependent A simulation study shows good behavior particular, apply ARFIMA Some real-life applications also illustrate approach.