Inequalities and limit theorems for random allocations

István Fazekas; Alexey Chuprunov; József Túri

Annales UMCS, Mathematica (2011)

  • Volume: 65, Issue: 1, page 69-85
  • ISSN: 2083-7402

Abstract

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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.

How to cite

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István Fazekas, Alexey Chuprunov, and József Túri. "Inequalities and limit theorems for random allocations." Annales UMCS, Mathematica 65.1 (2011): 69-85. <http://eudml.org/doc/267820>.

@article{IstvánFazekas2011,
abstract = {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.},
author = {István Fazekas, Alexey Chuprunov, József Túri},
journal = {Annales UMCS, Mathematica},
keywords = {Random allocation; moment inequality; merge theorem; almost sure limit theorem; random allocation},
language = {eng},
number = {1},
pages = {69-85},
title = {Inequalities and limit theorems for random allocations},
url = {http://eudml.org/doc/267820},
volume = {65},
year = {2011},
}

TY - JOUR
AU - István Fazekas
AU - Alexey Chuprunov
AU - József Túri
TI - Inequalities and limit theorems for random allocations
JO - Annales UMCS, Mathematica
PY - 2011
VL - 65
IS - 1
SP - 69
EP - 85
AB - 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.
LA - eng
KW - Random allocation; moment inequality; merge theorem; almost sure limit theorem; random allocation
UR - http://eudml.org/doc/267820
ER -

References

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  1. 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.[WoS] Zbl1164.60013
  2. Békéssy, A., On classical occupancy problems. I, Magy. Tud. Akad. Mat. Kutató Int. Közl. 8 (1-2) (1963), 59-71. Zbl0126.34101
  3. 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. Zbl0979.60014
  4. Berkes, I., Csáki, E., A universal result in almost sure central limit theory, Stoch. Proc. Appl. 94(1) (2001), 105-134.[Crossref] Zbl1053.60022
  5. Chuprunov, A., Fazekas, I., Inequalities and strong laws of large numbers for random allocations, Acta Math. Hungar. 109, no. 1-2 (2005), 163-182.[WoS] Zbl1120.60300
  6. Fazekas, I., Chuprunov, A., Almost sure limit theorems for random allocations, Studia Sci. Math. Hungar. 42, no. 2 (2005), 173-194. Zbl1098.60029
  7. 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.[WoS] Zbl1134.60026
  8. Fazekas, I., Rychlik, Z., Almost sure functional limit theorems, Ann. Univ. Mariae Curie-Skłodowska Sect. A 56(1) (2002), 1-18. Zbl1050.60039
  9. Fazekas, I., Rychlik, Z., Almost sure central limit theorems for random fields, Math. Nachr. 259 (2003), 12-18. Zbl1028.60019
  10. Hórmann, S., An extension of almost sure central limit theory, Statist. Probab. Lett. 76, no. 2 (2006), 191-202. Zbl1087.60509
  11. 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. 
  12. Kolchin, V. F., Sevast'yanov, B. A. and Chistyakov, V. P., Random Allocations, V. H. Winston & Sons, Washington D. C., 1978. 
  13. Matuła, P., On almost sure limit theorems for positively dependent random variables, Statist. Probab. Lett. 74, no. 1 (2005), 59-66. Zbl1080.60026
  14. Rényi, A., Three new proofs and generalization of a theorem of Irving Weiss, Magy. Tud. Akad. Mat. Kutató Int. Közl. 7(1-2) (1962), 203-214. Zbl0113.12704
  15. Orzóg, M., Rychlik, Z., On the random functional central limit theorems with almost sure convergence, Probab. Math. Statist. 27, no. 1 (2007), 125-138. Zbl1131.60022
  16. Weiss, I., Limiting distributions in some occupancy problems, Ann. Math. Statist. 29(3) (1958), 878-884.[Crossref] Zbl0088.11304

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