On the Poisson approximation of the binomial distribution (Q2773622)

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\[NEWLINE \rho(z,P,Q) > \exp\Biggl\{-4\sqrt{nz\Bigl(\log \frac{z}{np^2}+4\Bigr)^3}\Biggr\} \quad \text{ if } p\leq\frac{1}{10} \;\text{ and } np^2\leq z\leq\frac{n}{10\log\frac{1}{p}} .NEWLINE\]