Randomized divisible statistics in a generalized distribution scheme with respect to a denumerable set of cells. (Russian) Zbl 0762.62007

Mirakhmedov, Sh. A.

Diskretn. Mat. 1, No. 4, 46-62 (1989).

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

Keywords:

asymptotic normality; sequence of independent and integer-valued nonnegative random variables; conditional distribution; Randomized decomposable statistics; central limit theorem; rate of convergence; local limit theorem