ON TYPES OF LIMIT DISTRIBUTIONS FOR SOME CLASS OF STATISTICS

Fbr statistical applications it is necessary to investigate the limit distributions of the functions of statistics based on a random sample from a general population (cf. [1], [2], [3]). All of these authors consider this problem under the condition that the distribution function of the population random variable does not depend on the sample size JV. For the first time Zubkov ([0]) has given sufficient conditions for the conver-gence to the normal and X2 distributions, in the case of a concrete function of the parameters (statistical estimation of entropy in the multinomial scheme), under the assumption that the sampling probabilities depend on N. We shall now determine the class of all the limit distributions of the distribu-tion of a function.t/(.) depending on the frequencies of the occurences of outcomes in a multinomial scheme when the probabilities of the outcomes may depend on /v. Let fi;, i: 7r..., s, be the frequencies of the occurences of the outcomes in a multinomial scheme, p; = pi(N) the true probabilities of the outcomes satisfying the conditions: the limits in (1) and (2) exist and have the following properties (1) nl!5ri(N):19, pl,>...>p2, i:1,...,s, JgIrp,;(N), J' * Np, Let us consider a function fl(.) on an s-dimensional simplex,S and let us define the function fr(.) Ut nfil H(fr) for t (*t,..., rr), fr1-1- g- (*rt...,fi"). Let us assume that fr (.) satisfies the followirtg conditions