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Randomized divisible statistics in a generalized distribution scheme with respect to a denumerable set of cells. (Russian) Zbl 0762.62007
Let $$\xi(n)=(\xi_ 1(n),\xi_ 2(n),\cdots)$$ be a sequence of independent and integer-valued nonnegative random variables depending on a parameter $$n\in\{1,2,\cdots\}$$. Assume that the series $$\zeta(n)=\sum^ \infty_{m=1}\xi_ m(n)$$ converges with probability 1 and that $$P\{\zeta(n)=n\}>0$$ for each $$n$$. The author considers a random vector $$\eta(n)=(\eta_ 1(n),\eta_ 2(n),\cdots)$$ whose distribution coincides with the conditional distribution of $$\xi(n)$$ given $$\zeta(n)=n$$. Randomized decomposable statistics are defined by $$R(\eta(n))=\sum^ \infty_{m=1}f_ m^{(n)}(\eta_ m(n))$$, where $$f_ 1^{(n)}(y)$$, $$f_ 2^{(n)}(y),\cdots$$ are random functions and $$y\in\{0,1,\cdots\}$$.
In the present paper a central limit theorem for such statistics is proved as $$n\to\infty$$ and an estimate of the rate of convergence in it is obtained. Moreover, the author gives sufficient conditions for sums of the type of $$\zeta(n)$$ which imply a convergence to the standard normal distribution in the form of a local limit theorem.

##### MSC:
 62E20 Asymptotic distribution theory in statistics 60F05 Central limit and other weak theorems