Limit theorems for iterates of the Sz\'asz-Mirakyan operator in probabilistic view

TL;DR

This paper investigates the long-term behavior of the iterates of the Szász-Mirakyan operator, revealing their convergence to a diffusion process and providing a probabilistic interpretation of this limit.

Contribution

It establishes the convergence of the iterates to a semigroup generated by a degenerate differential operator and offers a probabilistic perspective linking Markov chains and diffusion processes.

Findings

Iterates converge uniformly to a continuous semigroup.

The limit process is characterized as a diffusion on the half line.

A probabilistic interpretation via Markov chains is provided.

Abstract

The Sz\'asz-Mirakyan operator is known as a positive linear operator which uniformly approximates a certain class of continuous functions on the half line. The purpose of the present paper is to find out limiting behaviors of the iterates of the Sz\'asz-Mirakyan operator in a probabilistic point of view. We show that the iterates of the Sz\'asz-Mirakyan operator uniformly converges to a continuous semigroup generated by a second order degenerate differential operator. A probabilistic interpretation of the convergence in terms of a discrete Markov chain constructed from the iterates and a limiting diffusion process on the half line is captured as well.