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## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2020 | 74 | 1 |
Tytuł artykułu

### On the number of empty cells in the allocation scheme of indistinguishable particles

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EN
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EN
The allocation scheme of $$n$$ indistinguishable particles into $$N$$ different cells is studied. Let the random variable $$\mu_0(n,K,N)$$ be the number of empty cells among the first $$K$$ cells. Let $$p=\frac{n}{n+N}$$. It is proved that $$\frac{\mu_0(n,K,N)-K(1-p)}{\sqrt{ K p(1-p)}}$$ converges in distribution to the Gaussian distribution with expectation zero and variance one, when $$n,K, N\to\infty$$ such that $$\frac{n}{N}\to\infty$$ and $$\frac{n}{NK}\to 0$$. If $$n,K, N\to\infty$$ so that $$\frac{n}{N}\to\infty$$ and $$\frac{NK}{n}\to \lambda$$, where $$0<\lambda<\infty$$, then $$\mu_0(n,K,N)$$ converges in distribution to the Poisson distribution with parameter $$\lambda$$. 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.
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EN
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wydano
2020
online
2020-10-20
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autor
Bibliografia
• Barbour, A. D., Holst, L., Janson, S., Poisson Approximation, Oxford University Press, Oxford, 1992.
• Chuprunov, A. N., Fazekas, I., Poisson limit theorems for the generalized allocation scheme, Ann. Univ. Sci. Budapest, Sect. Comp. 49 (2019), 77–96.
• Gibbons, J. D., Nonparametric Statistical Inference, McGraw-Hill, New York, 1971.
• Gordon, L., Schilling, M. F., Waterman, M. S., An extreme value theory for long head runs, Probab. Theory Related Fields 72 (1986), 279–287.
• 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.
• Kolchin, V. F., A class of limit theorems for conditional distributions, Litovsk. Mat. Sb. 8 (1968), 53–63 (Russian).
• Kolchin, V. F., Random Graphs, Cambridge University Press, Cambridge, 1999.
• Kolchin, V. F., Sevast’yanov, B. A., Chistyakov, V. P., Random Allocations, V. H. Winston & Sons, Washington D. C., 1978.
• Renyi, A., Probability Theory, Elsevier, New York, 1970.
• Timashev, A. N., Asymptotic Expansions in Probabilistic Combinatorics, TVP Science Publishers, Moscow, 2011 (Russian).
• Trunov, A. N., Limit theorems in the problem of distributing identical particles in different cells, Proc. Steklov Inst. Math. 177 (1988), 157–175.
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