On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell

Abstract

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 K cells. We prove that, under some conditions, this K-dimensional random vector with centralised and normalised coordinates converges in distribution to the K-dimensional standard Gaussian law. We obtain both local and integral versions of this limit theorem. The above limit theorem implies a χ2 limit theorem which leads to a χ2-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 χ2-test. To handle the mathematical problem, we insert our model into the framework of Kolchin’s generalized allocation scheme.

Keywords

• random allocation
• generalized allocation scheme
• Poisson distribution
• Gaussian distribution
• limit theorem
• local limit theorem