Geodesic
On the number of empty cells in the allocation scheme of indistinguishable particles
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 74 (2020) no. 1.

Voir la notice de l'article provenant de la source Library of Science

The allocation scheme of n indistinguishable particles into N different cells is studied. Let the random variable μ_0(n,K,N) be the number of empty cells among the first K cells. Let p=n/n+N. It is proved that μ_0(n,K,N)-K(1-p)/√( K p(1-p)) converges in distribution to the Gaussian distribution with expectation zero and variance one, when n,K, N→∞ such that n/N→∞ and n/NK→ 0. If n,K, N→∞ so that n/N→∞ and NK/n→λ, where 0∞, then μ_0(n,K,N) converges in distribution to the Poisson distribution with parameter λ. Two applications of the results are given to mathematical statistics. First, a method  is offered to test the value of n. Then, an analogue of the run-test is suggested with an application in signal processing.
Keywords: Allocation scheme of indistinguishable particles into different cells, Gaussian random variable, Berry-Esseen inequality, limit theorem, local limit theorem 
@article{AUM_2020_74_1_a6,
     author = {Chuprunov, Alexey and Fazekas, Istvan},
     title = {On the number of empty cells in the allocation scheme of indistinguishable particles},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {74},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a6/}
}
TY  - JOUR
AU  - Chuprunov, Alexey
AU  - Fazekas, Istvan
TI  - On the number of empty cells in the allocation scheme of indistinguishable particles
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2020
VL  - 74
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a6/
LA  - en
ID  - AUM_2020_74_1_a6
ER  - 
%0 Journal Article
%A Chuprunov, Alexey
%A Fazekas, Istvan
%T On the number of empty cells in the allocation scheme of indistinguishable particles
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2020
%V 74
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a6/
%G en
%F AUM_2020_74_1_a6
Chuprunov, Alexey; Fazekas, Istvan. On the number of empty cells in the allocation scheme of indistinguishable particles. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 74 (2020) no. 1. http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a6/

[1] Barbour, A. D., Holst, L., Janson, S., Poisson Approximation, Oxford University Press, Oxford, 1992.

[2] Chuprunov, A. N., Fazekas, I., Poisson limit theorems for the generalized allocation scheme, Ann. Univ. Sci. Budapest, Sect. Comp. 49 (2019), 77–96.

[3] Gibbons, J. D., Nonparametric Statistical Inference, McGraw-Hill, New York, 1971.

[4] Gordon, L., Schilling, M. F., Waterman, M. S., An extreme value theory for long head runs, Probab. Theory Related Fields 72 (1986), 279–287.

[5] Khakimullin, E. R., Enatskaya, N. Yu., Limit theorems for the number of empty cells, Diskret. Mat. 9 (2) (1997), 120–130 (Russian); translation in Discrete Math. Appl. 7 (2) (1997), 209–219.

[6] Kolchin, V. F., A class of limit theorems for conditional distributions, Litovsk. Mat. Sb. 8 (1968), 53–63 (Russian).

[7] Kolchin, V. F., Random Graphs, Cambridge University Press, Cambridge, 1999.

[8] Kolchin, V. F., Sevast’yanov, B. A., Chistyakov, V. P., Random Allocations, V. H. Winston Sons, Washington D. C., 1978.

[9] Renyi, A., Probability Theory, Elsevier, New York, 1970.

[10] Timashev, A. N., Asymptotic Expansions in Probabilistic Combinatorics, TVP Science Publishers, Moscow, 2011 (Russian).

[11] Trunov, A. N., Limit theorems in the problem of distributing identical particles in different cells, Proc. Steklov Inst. Math. 177 (1988), 157–175.