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On a number of particles in a marked set of cells in a general allocation scheme

  • Alexey N. Chuprunov EMAIL logo
Published/Copyright: June 18, 2023
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 33 Issue 3

Abstract

In a generalized allocation scheme of n particles over N cells we consider the random variable ηn,N(K) which is the number of particles in a given set consisting of K cells. We prove that if n, K, N → ∞, then under some conditions random variables ηn,N(K) are asymptotically normal, and under another conditions ηn,N(K) converge in distribution to a Poisson random variable. For the case when N → ∞ and n is a fixed number, we find conditions under which ηn,N(K) converge in distribution to a binomial random variable with parameters n and s = KN , 0 < K < N, multiplied by a integer coefficient. It is shown that if for a generalized allocation scheme of n particles over N cells with random variables having a power series distribution defined by the function B(β) = ln(1 − β) the conditions n, N, K → ∞, KN → s, N = γ ln(n) + o(ln(n)), where 0 < s < 1, 0 < γ < ∞, are satisfied, then distributions of random variables ηn,N(K)n converge to a beta-distribution with parameters sγ and (1 − s)γ.

Keywords: generalized allocation scheme; Poisson distribution; Gaussian distribution; binomial distribution; hypergeometric distribution; beta-distribution; local limit theorem

Originally published in Diskretnaya Matematika (2022) 34, №3, 141–152 (in Russian).

References

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Received: 2021-08-27
Published Online: 2023-06-18
Published in Print: 2023-06-27

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On scatter properties of modular addition operation over imprimitivity systems of the translation group of the binary vector space
  3. On a number of particles in a marked set of cells in a general allocation scheme
  4. On the equality problem of finitely generated classes of exponentially-polynomial functions
  5. Fault-tolerant resolvability of some graphs of convex polytopes
  6. On bases of all closed classes of Boolean vector functions
  7. 10.1515/dma-2023-0018
https://doi.org/10.1515/dma-2023-0014
Keywords for this article
generalized allocation scheme; Poisson distribution; Gaussian distribution; binomial distribution; hypergeometric distribution; beta-distribution; local limit theorem

Articles in the same Issue

  1. Frontmatter
  2. On scatter properties of modular addition operation over imprimitivity systems of the translation group of the binary vector space
  3. On a number of particles in a marked set of cells in a general allocation scheme
  4. On the equality problem of finitely generated classes of exponentially-polynomial functions
  5. Fault-tolerant resolvability of some graphs of convex polytopes
  6. On bases of all closed classes of Boolean vector functions
  7. 10.1515/dma-2023-0018
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