  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.

Probability;Functional Analysis