Universal codes as a basis for nonparametric testing of serial independence for time series

We address the problem of nonparametric testing of serial independence for time series and its generalization. More precisely, we consider a stationary and ergodic source p, which generates symbols x1…xt from some finite set A and a null hypothesis H0 that p is a Markov source of order at most m,(m⩾0). The alternative hypothesis H1 is that the sequence is generated by a stationary and ergodic source, which differs from the source under H0. In particular, if m=0 we have the null hypothesis H0 that the sequence is generated by a Bernoulli source (i.e. the hypothesis that x1…xt are independent). In this paper some new tests that are based on so-called universal codes and universal predictors, are suggested.