In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimators with Poisson weights on the positive half-line (also called Szasz estimators).

翻译：在本文中,我们证明Poisson分布与Gaussian分布之比的局部限值理论,其平均值和差异相同,只使用基本方法(Taylor扩张和Stirling的公式 ) 。 然后,我们应用结果在Poisson和Gaussian实验之间的Le Cam距离上得出一个上限,这充分证明了Pollard(2010年)未发表的一套演讲说明中提供的草图,而Pollarard(2010年)采用了不同的方法。我们还使用本地限值来得出Bernstein c.d.f.和以Poisson重量表示正中线(也称为Szasz估计器)的密度测算器。