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The Sz\'asz-Mirakyan operator is known as a positive linear operator whichuniformly approximates a certain class of continuous functions on the halfline. The purpose of the present paper is to find out limiting behaviors of theiterates of the Sz\'asz-Mirakyan operator in a probabilistic point of view. Weshow that the iterates of the Sz\'asz-Mirakyan operator uniformly converges toa continuous semigroup generated by a second order degenerate differentialoperator. A probabilistic interpretation of the convergence in terms of adiscrete Markov chain constructed from the iterates and a limiting diffusionprocess on the half line is captured as well.

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