A local limit theorem for the Poisson distribution and its application to the Le Cam distance between Poisson and Gaussian experiments and asymptotic properties of Szasz estimators

In this paper, we develop a precise local limit theorem for the Poisson distribution. We then apply the result to prove an upper bound on the Le Cam distance between Poisson and Gaussian experiments. We also use it 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).