On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell
We consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first <i>K</i> cells. We prove that, under some cond...
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MDPI AG
2022-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/7/1099 |
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| author | Alexey Nikolaevich Chuprunov István Fazekas |
| author_facet | Alexey Nikolaevich Chuprunov István Fazekas |
| author_sort | Alexey Nikolaevich Chuprunov |
| collection | DOAJ |
| description | We consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first <i>K</i> cells. We prove that, under some conditions, this <i>K</i>-dimensional random vector with centralised and normalised coordinates converges in distribution to the <i>K</i>-dimensional standard Gaussian law. We obtain both local and integral versions of this limit theorem. The above limit theorem implies a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula> limit theorem which leads to a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula>-test. The parity bit method does not detect even numbers of errors in binary files; therefore, our model can be applied to describe the distribution of errors in those files. For the homogeneous allocation model, we obtain a limit theorem when both the number of particles and the number of cells tend to infinity. In that case, we prove convergence to the finite dimensional distributions of the Brownian bridge. This result also implies a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula>-test. To handle the mathematical problem, we insert our model into the framework of Kolchin’s generalized allocation scheme. |
| first_indexed | 2024-03-09T11:37:44Z |
| format | Article |
| id | doaj.art-055397a011a64d7d9a124eec362916c2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:37:44Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-055397a011a64d7d9a124eec362916c22023-11-30T23:37:10ZengMDPI AGMathematics2227-73902022-03-01107109910.3390/math10071099On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each CellAlexey Nikolaevich Chuprunov0István Fazekas1Faculty of Applied Mathematics, Physics and Information Technology, Chuvash State University, Universitetskaia Str. 38, 428015 Cheboksary, RussiaFaculty of Informatics, University of Debrecen, Egyetem Square 1, 4032 Debrecen, HungaryWe consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first <i>K</i> cells. We prove that, under some conditions, this <i>K</i>-dimensional random vector with centralised and normalised coordinates converges in distribution to the <i>K</i>-dimensional standard Gaussian law. We obtain both local and integral versions of this limit theorem. The above limit theorem implies a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula> limit theorem which leads to a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula>-test. The parity bit method does not detect even numbers of errors in binary files; therefore, our model can be applied to describe the distribution of errors in those files. For the homogeneous allocation model, we obtain a limit theorem when both the number of particles and the number of cells tend to infinity. In that case, we prove convergence to the finite dimensional distributions of the Brownian bridge. This result also implies a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula>-test. To handle the mathematical problem, we insert our model into the framework of Kolchin’s generalized allocation scheme.https://www.mdpi.com/2227-7390/10/7/1099random allocationgeneralized allocation schemePoisson distributionGaussian distributionlimit theoremlocal limit theorem |
| spellingShingle | Alexey Nikolaevich Chuprunov István Fazekas On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell Mathematics random allocation generalized allocation scheme Poisson distribution Gaussian distribution limit theorem local limit theorem |
| title | On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell |
| title_full | On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell |
| title_fullStr | On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell |
| title_full_unstemmed | On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell |
| title_short | On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell |
| title_sort | on the numbers of particles in cells in an allocation scheme having an even number of particles in each cell |
| topic | random allocation generalized allocation scheme Poisson distribution Gaussian distribution limit theorem local limit theorem |
| url | https://www.mdpi.com/2227-7390/10/7/1099 |
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