-- The asymptotic behaviour of the number of absent s-chains in a polynomial scheme is studied. We consider the case where the length of chains s is constant, s > 2, and the number of trials, the number of outcomes and their probabilities are related by the conditions determining an analogue of the central domain. Some asymptotic expressions for the mean and variance are given and the asymptotic normality is proved. 1. INTRODUCTION In this paper some topics concerning the asymptotic behaviour of the number of absent s-chains in a polynomial scheme are studied. We consider the case where the length of chains is constant, the number of trials and the number of outcomes of the scheme tend to infinity in such a way that 0 < < -- = < 2 < oo, and the probabilities of outcomes pi,...,PN vary in such a way that 0<NPk 2. In this case the statistic is a decomposable statistic on the frequencies of outcomes of the Markov chain generated by s-chains. It seems to us that the time is appropriate to extend the investigations of such statistics since in the case of the proper polynomial scheme, i.e., in the case
Discrete Mathematics and Applications – de Gruyter
Published: Jan 1, 1993
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