Poisson approximation to the binomial distribution: extensions to the convergence of positive operators

TL;DR

This paper explores Poisson approximation to the binomial distribution and applies it to analyze the convergence of positive linear operators, using analytic methods to recover and extend previous probabilistic results.

Contribution

It introduces an analytic approach to Poisson approximation for positive operators, enlarging the class of limit operators and providing new characterizations.

Findings

Recovered known convergence results using analytic methods

Extended the list of limit operators in Poisson approximation

Provided new characterizations of limit operators

Abstract

The idea behind Poisson approximation to the binomial distribution was used in [J. de la Cal, F. Luquin, J. Approx. Theory, 68(3), 1992, 322-329] and subsequent papers in order to establish the convergence of suitable sequences of positive linear operators. The proofs in these papers are given using probabilistic methods. We use similar methods, but in analytic terms. In this way we recover some known results and establish several new ones. In particular, we enlarge the list of the limit operators and give characterizations of them.