Randomized divisible statistics in a scheme of independent allocation of particles on cells. (Russian) Zbl 0711.60011

Mirakhmedov, Sh. A.

Diskretn. Mat. 2, No. 2, 97-111 (1990).

n particles are distributed independently of each other into infinitely many cells numbered by 1,2,.... A particle hits the m th cell with probability \(p_ m\geq 0\), \(\sum^{\infty}_{m=1}p_ m=1\). Let \(\eta_ m=\eta_ m(n)\) denote the number of particles thrown into the m th cell, and let \(\{f_ m^{(n)}(y)\), \(m=1,2,...\}\) be a sequence of random functions of the non-negative and integer argument y. Randomized divisible statistics R(\(\eta\)) are defined by \[ R(\eta)=\sum^{\infty}_{m=1}f_ m^{(n)}(y), \] assuming that this series converges with probability 1. The author considers various applications of his previous results [see ibid. 1, No.4, 46-62 (1989)] concerning the asymptotic behaviour of R(\(\eta\)) as \(n\to \infty\).

Reviewer: L.Mutafchiev

MSC:

60C05	Combinatorial probability
62F05	Asymptotic properties of parametric tests
60F05	Central limit and other weak theorems

Keywords:

allocation of particles on cells; infinitely many cells; Randomized divisible statistics; asymptotic behaviour