On the Poisson approximation of the binomial distribution
Given two arbitrary probability distributions \(P\) and \(Q\) on the real line and an arbitrary nonnegative constant \(z\), denote by \(\rho(z,P,Q)\) the so-called Dudley distance between \(P\) and \(Q\): NEWLINE\[NEWLINE \rho(z,P,Q)=\inf_{\xi,\eta}\mathbf{P}\{|\xi-\eta|>z\},NEWLINE\]NEWLINE where the infimum is calculated over all random variables \(\xi\) and \(\eta\) on a common probability space with distributions \(P\) and \(Q\), respectively. Let \(P\) be the binomial distribution with parameters \(n\) and \(p\) and let \(Q\) be the Poisson distribution with parameter \(\lambda=np\). The main result of the article states that for all \(p\leq 1/2\) and integer \(z\geq 1\) the following estimates hold: NEWLINE\[NEWLINE \begin{aligned} \rho(z,P,Q) < \frac{7}{3} \exp\Bigl\{-\frac{1}{27}\frac{z^2}{np^3}\Bigr\} \quad &\text{ for } z\leq np^2, \\ \rho(z,P,Q) < \exp\left\{-\frac{1}{15}np\right\} \quad &\text{ for } z\geq np^2, \\ \rho(z,P,Q) < \exp\Biggl\{-\frac{1}{4}\sqrt{nz\Bigl(\log \frac{z}{np^2}-2\Bigr)^3}\Biggr\} \quad &\text{ for } e^4np^2\leq z\leq \frac{n(1-p)}{\log\frac{1}{p}}. \end{aligned} NEWLINE\]NEWLINE Moreover, NEWLINE\[NEWLINE \rho(z,P,Q) > \exp\Biggl\{-4\sqrt{nz\Bigl(\log \frac{z}{np^2}+4\Bigr)^3}\Biggr\} \quad \text{ if } p\leq\frac{1}{10} \;\text{ and } np^2\leq z\leq\frac{n}{10\log\frac{1}{p}} .NEWLINE\]
scientific article; zbMATH DE number 1710250