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We consider a tuple of states of an ( s  – 1)-order Markov chain whose transition probabilities depend on a small part of  s  – 1 preceding states. We obtain limit distributions of certain χ 2 -statistics  X  and  Y  based on frequencies of tuples of states of the Markov chain. For the statistic  X , frequencies of tuples of only those states are used on which the transition probabilities depend, and for the statistic  Y , frequencies of  s -tuples without gaps. The statistical test with statistic  X  which distinguishes the hypotheses  H  1  (a high-order Markov chain) and  H  0  (an independent equiprobable sequence) appears to be more powerful than the test with statistic  Y  . The statistic  Z  of the Neyman–Pearson test, as well as  X , depends only on frequencies of tuples with gaps. The statistics  X  and  Y  are calculated without use of distribution parameters under the hypothesis  H  1 , and their probabilities of errors of the first and second kinds depend only on the non-centrality parameter, which is a function of transition probabilities. Thus, for these statistics the  hypothesis  H  1  can be considered as composite.

On two statistics of chi-square type based on frequencies of tuples of states of a high-order Markov chain