A chi-square statistic based on the chains of outcomes of independent trials with gaps

The detection of dependences in random sequences in the cases, where the dependence makes itself evident only on frequencies of long tuples of outcomes, involves difficulties connected with bounded possibilities of computers. In this case, the use is made of different criteria based on statistics depending on long tuples of successive symbols in the sequence under consideration or on incomplete tuples of symbols separated from each other. The criteria of such type can be found in [1]. In this paper, we find the limit distributions of chi-square statistics based on incomplete tuples of outcomes of independent trials in the case, where the number of trials tends to infinity, the outcomes are equiprobable, and the number of outcomes is finite.
This research was supported by the Russian Foundation of Basic Research, grant 99–01–00012, and by the Council for State Support of Leading Scientific Schools, grant 00–15–96136.