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

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