Calculate central limit theorem for the number of empty cells after allocation of particles

This work belongs to the field of limit theorems for separable statistics. In particular, this paper considers the number of empty cells after placing particles in a finite number of cells, where each particle is placed in a polynomial scheme. The statistics under consideration belong to the class of separable statistics, which were previously considered in (Mirakhmedov: 1985), where necessary statements for the layout of particles in a countable number of cells were proved. The same scheme was considered in (Asimov: 1982), in which the conditions for the asymptotic normality of random variables were established. In this paper, the asymptotic normality of the statistics in question is proved and an estimate of the remainder term in the central limit theorem is obtained. In summary, the demand for separable statistics systems is growing day by day to address large-scale databases or to facilitate user access to data management. Because such systems are not only used for data entry and storage, they also describe their structure: file collection supports logical consistency; provides data processing language; restores data after various interruptions; database management systems allow multiple users.