On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

TL;DR

This paper establishes a local limit theorem for Poisson to Gaussian ratios, derives bounds on their experiment distances, and analyzes the asymptotic variance of Szasz estimators, providing new elementary proofs and correcting previous errors.

Contribution

It introduces an elementary proof of the local limit theorem, bounds the Le Cam distance between Poisson and Gaussian experiments, and studies the asymptotics of Szasz estimators' variance.

Findings

Proved a local limit theorem using elementary methods.

Derived an upper bound on the Le Cam distance.

Established asymptotics for Szasz estimators' variance.

Abstract

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Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach.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.