Calculate central limit theorem for the number of empty cells after allocation of particles
Keywords:
central limit theorem, particle placement, particle, polynomial scheme, separable statisticsAbstract
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.
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Ardell, A. J. (1972). The effect of volume fraction on particle coarsening: theoretical considerations. Acta metallurgica, 20(1), 61-71. https://doi.org/10.1016/0001-6160(72)90114-9
Asimov, I. (1982). Exploring the earth and the cosmos: the growth and future of human knowledge. New York: Crown.
Benedicta, O. (2021). Relationship between competitive intelligence and competitive advantage in manufacturing industry. International Research Journal of Management, IT and Social Sciences, 8(5), 342-351. https://doi.org/10.21744/irjmis.v8n5.1908
Cesati, M., & Trevisan, L. (1997). On the efficiency of polynomial time approximation schemes. Information Processing Letters, 64(4), 165-171. https://doi.org/10.1016/S0020-0190(97)00164-6
Chaloupka, V., Barbaro-Galtieri, A., Chew, D. M., Kelly, R. L., Lasinski, T. A., Rittenberg, A., ... & Particle Data Group. (1974). Review of particle properties. Physics Letters B, 50(1), i. https://doi.org/10.1016/0370-2693(74)90738-2
Dedecker, J., & Rio, E. (2000, January). On the functional central limit theorem for stationary processes. In Annales de l'Institut Henri Poincare (B) Probability and Statistics (Vol. 36, No. 1, pp. 1-34). No longer published by Elsevier. https://doi.org/10.1016/S0246-0203(00)00111-4
Degond, P., Peyrard, P. F., Russo, G., & Villedieu, P. (1999). Polynomial upwind schemes for hyperbolic systems. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(6), 479-483. https://doi.org/10.1016/S0764-4442(99)80194-3
El-Zonkoly, A. M. (2011). Optimal placement of multi-distributed generation units including different load models using particle swarm optimization. Swarm and Evolutionary Computation, 1(1), 50-59. https://doi.org/10.1016/j.swevo.2011.02.003
Hall, P. (1984). Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of multivariate analysis, 14(1), 1-16. https://doi.org/10.1016/0047-259X(84)90044-7
Indahyati, N., & Sintaasih, D. K. (2019). The relationship between organizational justice with job satisfaction and organizational citizenship behavior. International Research Journal of Management, IT and Social Sciences, 6(2), 63-71. https://doi.org/10.21744/irjmis.v6n2.611
Ivchenko, G. I., & Levin, V. V. (1978). Asymptotic normality in the scheme of simple random sampling without replacement. Theory of Probability & Its Applications, 23(1), 93-105.
Kolchin, V. F., Sevastianov, B. A., & Chistyakov, V. P. (1976). Random accommodations. Science, Moscow.
Lee, C. S., Ayala, H. V. H., & dos Santos Coelho, L. (2015). Capacitor placement of distribution systems using particle swarm optimization approaches. International Journal of Electrical Power & Energy Systems, 64, 839-851. https://doi.org/10.1016/j.ijepes.2014.07.069
Mikhailov, V. G. (1981). The central limit theorem for a scheme of independent allocation of particles by cells. Trudy Matematicheskogo Instituta imeni VA Steklova, 157, 138-152.
Mirakhmedov, S. A. (1987). Estimates for the convergence rate in the central limit theorem for randomized separable statistics in a multinomial scheme. Journal of Soviet Mathematics, 38(6), 2346-2357.
Mirakhmedov, S. A. (1988). Approximation of the Distribution of Multidimensional Randomized Divisible Statistics by Normal Distributions (Multinomial Scheme). Theory of Probability & Its Applications, 32(4), 696-706.
Mirakhmedov, S. A. (1989). Randomized decomposable statistics in a generalized allocation scheme over a countable set of cells. Diskretnaya matematika, 1(4), 46-62.
Mitchell, M. W., Genton, M. G., & Gumpertz, M. L. (2006). A likelihood ratio test for separability of covariances. Journal of Multivariate Analysis, 97(5), 1025-1043. https://doi.org/10.1016/j.jmva.2005.07.005
Popova, T. Y. (1968). Limit theorems in a model of distribution of particles of two types. Theory of Probability & Its Applications, 13(3), 511-516.
Rakhimov, B. S., Mekhmanov, M. S., & Bekchanov, B. G. (2021, April). Parallel algorithms for the creation of medical database. In Journal of Physics: Conference Series (Vol. 1889, No. 2, p. 022090). IOP Publishing.
Rakhimov, B., Rakhimova, F., Sobirova, S., & Allaberganov, O. (2021). Mathematical Bases Of Parallel Algorithms For The Creation Of Medical Databases. InterConf.
Valjarevi?, D., & Petrovi?, L. (2020). Statistical causality and separable processes. Statistics & Probability Letters, 167, 108915. https://doi.org/10.1016/j.spl.2020.108915
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