Limit theorems for iterates of the Sz\'asz-Mirakyan operator in probabilistic view
- Resource Type
- Working Paper
- Authors
- Akahori, Jirô; Namba, Ryuya; Semba, Shunsuke
- Source
- Journal of Theoretical Probability 36 (2023), 1321-1338
- Subject
- Mathematics - Probability
Mathematics - Functional Analysis
Primary 60J10, 60J60, Secondary 41A25, 41A36, 47D03
- Language
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.
Comment: 17 pages