Limit Theorems for Iterates of the Szász–Mirakyan Operator in Probabilistic View

Jiro Akahori, Ryuya Namba () and Shunsuke Semba ()
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Ryuya Namba: Shizuoka University
Shunsuke Semba: Ritsumeikan University

Journal of Theoretical Probability, 2023, vol. 36, issue 2, 1321-1338

Abstract: Abstract The Szász–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 Szász–Mirakyan operator in a probabilistic point of view. We show that the iterates of the Szász–Mirakyan operator uniformly converge 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.

Keywords: Bernstein operator; Szász–Mirakyan operator; Approximation of continuous functions; Diffusion process; Primary 60J10; 60J60; Secondary 41A25; 41A36; 47D03 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01199-5

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