Abstract

We consider an allocation scheme of \(2n\) distinguishable particles by \(N\) different cells under the condition than each cell contains an even number of particles. We show that this scheme is a general allocation scheme defined by the random variable \(\xi_{i}\) with the distribution \({\mathbf{P}}(\xi_{i}=2k)=\frac{\alpha^{2k}}{(2k)!\cosh\alpha},\)\(k=0,1,2\dots\). Let \(\mu_{2r}(N,K,n)\) be a number of cells from the first \(K\) cells that contain \(2r\) particles. We prove that under some types of convergence of \(n,K,N\) to infinity \(\mu_{2r}(N,K,n)\) converges in distribution to the Poisson random variable. The limit Poisson random variable is described.

中文翻译：

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每个单元中具有偶数个粒子的分配方案中的泊松极限定理

摘要

我们考虑了一个\（2n \）个可区分粒子的分配方案，该分配方案由\（N \）个不同的单元格组成，而不是每个单元格包含偶数个粒子。我们表明，这种方案是由随机变量所定义的一般分配方案\（\ xi_ {I} \）与分配\（{\ mathbf {P}}（\ xi_ {I} = 2K）= \压裂{\ alpha ^ {2k}} {（2k）！\ cosh \ alpha}，\）\（k = 0,1,2 \ dots \）。令\（\ mu_ {2r}（N，K，n）\）为包含\（2r \）粒子的前\（K \）个单元格中的多个单元格。我们证明在\（n，K，N \）收敛到无穷大\（\ mu_ {2r}（N，K，n）\）的某些类型下在分布上收敛到泊松随机变量。描述了极限泊松随机变量。