  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provided in the unpublished set of lecture notes by Pollard (2010), who uses a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.
