On some $\chi^2$-type statistics that depend functionally on estimators of unknown parameters

A series of theorems are known which state convergence of the distributions of $\chi^2$-type statistics based on frequencies of outcomes in a sequence of polynomial trials to some limit distribution. For chains of outcomes where gaps are present, similar results are known only in the case of equiprobable outcomes. In this paper, we extend these results to polynomial trials with arbitrary positive probabilities of outcomes; we also study statistics resulted from replacing the probabilities of outcomes in $\chi^2$-type statistics by their estimators. We obtain limit distributions of such statistics in the general case if no gaps are present, and in some special cases otherwise. We also give a limit theorem in the case where the sequence under consideration is a Markov chain which approaches polynomial trials.
This research was supported by the grant 1758.2003.1 of President of Russian Federation for support of leading scientific schools.