PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2020 | 74 | 1 |
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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.
Rocznik
Tom
74
Numer
1
Opis fizyczny
Daty
wydano
2020
online
2020-10-20
Twórcy
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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.ojs-doi-10_17951_a_2020_74_1_15-29
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.