Random allocations of balls into boxes are considered. Properties of the number of boxes containing a fixed number of balls are studied. A moment inequality is obtained. A merge theorem with Poissonian accompanying laws is proved. It implies an almost sure limit theorem with a mixture of Poissonian laws as limiting distribution. Almost sure versions of the central limit theorem are obtained when the parameters are in the central domain.
@article{bwmeta1.element.ojs-doi-10_17951_a_2011_65_1_69-85,
author = {Istvan Fazekas and Alexey Chuprunov and Jozsef Turi},
title = {Inequalities and limit theorems for random allocations},
journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
volume = {65},
year = {2011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_1_69-85}
}
Istvan Fazekas; Alexey Chuprunov; Jozsef Turi. Inequalities and limit theorems for random allocations. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_1_69-85/
Becker-Kern, P., An almost sure limit theorem for mixtures of domains in random allocation, Studia Sci. Math. Hungar. 44, no. 3 (2007), 331-354.
Bekessy, A., On classical occupancy problems. I, Magy. Tud. Akad. Mat. Kutató Int. Kozl. 8 (1-2) (1963), 59-71.
Berkes, I., Results and problems related to the pointwise central limit theorem, Szyszkowicz, B., (Ed.), Asymptotic Results in Probability and Statistics, Elsevier, Amsterdam, 1998, 59-96.
Berkes, I., Csaki, E., A universal result in almost sure central limit theory, Stoch. Proc. Appl. 94(1) (2001), 105-134.
Chuprunov, A., Fazekas, I., Inequalities and strong laws of large numbers for random allocations, Acta Math. Hungar. 109, no. 1-2 (2005), 163-182.
Fazekas, I., Chuprunov, A., Almost sure limit theorems for random allocations, Studia Sci. Math. Hungar. 42, no. 2 (2005), 173-194.
Fazekas, I., Chuprunov, A., An almost sure functional limit theorem for the domain of geometric partial attraction of semistable laws, J. Theoret. Probab. 20, no. 2 (2007), 339-353.
Fazekas, I., Rychlik, Z., Almost sure functional limit theorems, Ann. Univ. Mariae Curie-Skłodowska Sect. A 56(1) (2002), 1-18.
Fazekas, I., Rychlik, Z., Almost sure central limit theorems for random fields, Math. Nachr. 259 (2003), 12-18.
Hormann, S., An extension of almost sure central limit theory, Statist. Probab. Lett. 76, no. 2 (2006), 191-202.
Kolchin, A. V., Limit theorems for a generalized allocation scheme, Diskret. Mat. 15, no. 4 (2003), 148-157 (Russian); English translation in Discrete Math. Appl. 13, no. 6 (2003), 627-636.
Kolchin, V. F., Sevast’yanov, B. A. and Chistyakov, V. P., Random Allocations, V. H. Winston & Sons, Washington D. C., 1978.
Matuła, P., On almost sure limit theorems for positively dependent random variables, Statist. Probab. Lett. 74, no. 1 (2005), 59-66.
Renyi, A., Three new proofs and generalization of a theorem of Irving Weiss, Magy. Tud. Akad. Mat. Kutató Int. K¨ozl. 7(1-2) (1962), 203-214.
Orzóg, M., Rychlik, Z., On the random functional central limit theorems with almost sure convergence, Probab. Math. Statist. 27, no. 1 (2007), 125-138.
Weiss, I., Limiting distributions in some occupancy problems, Ann. Math. Statist. 29(3) (1958), 878-884.