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Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach.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). The propagation of errors in the literature due to the incorrect estimate in Lemma 2 (iv) of Leblanc (2012) is addressed in the Appendix.
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扩展和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)未发表的一套讲演说明中提供的草图,Pollarard(2010年)使用了不同的方法。我们还使用本地限值来得出Bernstein c.d.f.和以Poisson重量表示的密度估测器(也称为Szasz 估计器)在正线半线(也称为Szasz估计器)上的误差。附录中涉及Lemma 2 (iv) Leblanc(2012年)的误差。