## 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.

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## References

Altomare, F.: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory

**6**, 92–164 (2010)Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and Its Applications, de Gruyter Studies in Mathematics 17. Walter de Gruyter, Berlin (1994)

Altomare, F., Carbone, I.: On some degenerate differential operators on weighted function spaces. J. Math. Anal. Appl.

**213**, 308–333 (1997)Attalienti, A., Campiti, M.: Bernstein-type operators on the half line. Czechoslov. Math. J.

**52**, 851–860 (2002)Baskakov, V.A.: An instance of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk. SSSR (N. S.)

**113**, 249–251 (1957). (**in Russian**)Becker, M.: Global approximation theorems for Szász–Mirakjan and Baskakov operators in polynomial weight spaces. Indiana Univ. Math. J.

**27**, 127–142 (1978)Bernstein, S.N.:

*Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités*. Commun. Soc. Math. Kharkow**13**, 1–2 (1912–13)Bustamante, J.: Bernstein Operators and Their Properties, Birkhäuser. Springer, Cham (2017)

Campiti, M., Tacelli, C.: Rate of convergence in Trotter’s approximation theorem. Constr. Approx.

**28**, 333–341 (2008)Campiti, M., Tacelli, C.: Erratum to: rate of convergence in Trotter’s approximation theorem. Constr. Approx.

**31**, 459–462 (2010)Canonne, C.:

*A short note on Poisson tail bounds*, unpublished manuscript (2019). http://www.cs.columbia.edu/ccanonne/files/misc/2017-poissonconcentration.pdfEthier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)

Ethier, S.N., Kurtz, T.G.: Fleming-Viot processes in population genetics. SIAM J. Control Optim.

**31**, 345–386 (1993)Feller, W.:

*Diffusion processes in genetics*, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 227–246 (1950)Jachymski, J.: Convergence of iterates of linear operators and the Kelisky–Rivlin type theorems. Stud. Math.

**195**, 99–112 (2009)Kallenberg, O.: Fundations of Modern Probability, Second Edition, Probability and its Applications (New York). Springer, New York (2002)

Karlin, S., Ziegler, Z.: Iteration of positive linear operators. J. Approx. Theory

**3**, 310–339 (1970)Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pac. J. Math.

**21**, 511–520 (1967)Klenke, A.: Probability Theory, A Comprehensive Course, Universitext. Springer, London (2008)

Konstantopoulos, T., Yuan, L., Zazanis, M.A.: A fully stochastic approach to limit theorems for iterates of Bernstein operators. Expo. Math.

**36**, 143–165 (2018)Korovkin, P.P.: Convergence of linear positive operators in the spaces of continuous functions. Dokl. Akad. Nauk. SSSR (N. S.)

**90**, 961–964 (1953). (**in Russian**)Kurtz, T.G.: Extensions of Trotter’s operator semigroup approximation theorems. J. Funct. Anal.

**3**, 354–375 (1969)Mirakyan, G.: Approximation des Fonctions Continues au Moyen de Polynômes de la Forme \(e^{-nx}\sum _{k=0}^mC_{k, n}\chi _k\). Dokl. Akad. Nauk. SSSR

**31**, 201–205 (1941)Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)

Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand.

**45**, 239–244 (1950)Trotter, H.F.: Approximation of semi-groups of operators. Pac. J. Math.

**8**, 887–919 (1958)

## Acknowledgements

The second-named author is supported by JSPS KAKENHI Grant number 19K23410.

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## Appendices

### Appendix A. Trotter’s Approximation Theorem and Its Rate of Convergence

Trotter’s approximation theorem provides a sufficient condition that the iterates of a bounded linear operator acting on a Banach space converge to a \( C_0 \)-semigroup. We give the statement of Trotter’s approximation theorem when the iteration of a linear operator enjoys the contraction property.

### Proposition A.1

(cf. [26, Theorem 5.1], [22, Theorem 2.13]) Let \( (\mathcal {U}, \Vert \cdot \Vert _{\mathcal {U}}) \) be a Banach space. Suppose that a sequence \( \{T_n\}_{n=1}^\infty \) of bounded linear operators on \( \mathcal {U} \) satisfies that \( \Vert T_n\Vert \le 1 \) for \( n \in \mathbb {N}\). Put \( \mathcal {A}_n:=n(T_n-I), \, n \in \mathbb {N}. \) We define a linear operator \( \mathcal {A} \) by the closure of the limit of \( \mathcal {A}_n \). If the domain \( D(\mathcal {A}) \) is dense in \( \mathcal {U} \) and the range of \( \lambda - \mathcal {A} \) is dense in \( \mathcal {U} \) for some \( \lambda >0 \), then there exists a \( C_0 \)-semigroup \( ({\mathsf {T}}_t)_{t \ge 0} \) acting on \( \mathcal {U} \) satisfying \(\Vert {\mathsf {T}}_t\Vert \le 1, \, t \ge 0\), and

On the other hand, as is easily seen, Proposition A.1 does not imply any quantitative estimates of (A.1). Campiti and Tacelli established in [9] a refinement of Proposition A.1 by giving the rate of convergence of (A.1).

### Proposition A.2

(cf. [9, Theorem 1.1], see also [10]) Suppose that \( \mathcal {U} \), \( \{T_n\}_{n=1}^\infty \), \( \{\mathcal {A}_n\}_{n=1}^\infty \) and \( ({\mathsf {T}}_t)_{t \ge 0} \) are as in Proposition A.1. Let \( \mathcal {D} \) be a dense subspace of \( \mathcal {U} \). We assume that

where \( \varphi _n, \psi _n : \mathcal {D} \rightarrow [0, \infty ) \) are semi-norms with \( \psi _n(f) \rightarrow 0 \) as \( n \rightarrow \infty \) for \( f \in \mathcal {D} \). Then, for \( t \ge 0 \) and \( f \in \{g \in \mathcal {D} \, | \, {\mathsf {T}}_t g \in \mathcal {D}, \, t \ge 0\} \), we have

Indeed, we have used Proposition A.2 in order to deduce the rate of convergence of the iterates of the Szász–Mirakyan operator (see Theorem 1.5).

### Appendix B. Korovkin-Type Theorems

It is well-known that the Korovkin-type theorems provide a quite powerful sufficient condition to deduce that a sequence of positive linear operators acting on some function spaces converges strongly to the identity operator. Korovkin’s first theorem, which was first discovered by Korovkin himself in [21], is stated as follows:

### Proposition B.1

(Korovkin’s first theorem) Let \( e_0(x) \equiv 1 \), \( e_1(x)=x \) and \( e_2(x)=x^2 \) for \( x \in [0, 1] \). If a sequence \( \{L_n\}_{n=1}^\infty \) of positive linear operators acting on *C*([0, 1]) satisfies

then, it holds that

So far, a number of generalizations of Proposition B.1 have been established in various settings. We refer to [1] and [2] for good surveys of this topic. In the sequel, \( (\mathfrak {X}, \Vert \cdot \Vert _{\mathfrak {X}}) \) is used in order to represent Banach spaces of some continuous functions on a locally compact space \(E \subset \mathbb {R}\). We give a general formulation of the Korovkin-type theorem.

### Definition B.2

*(Korovkin set)* A subset \(\mathcal {K}\) of \(\mathfrak {X}\) is called a Korovkin set of \(\mathfrak {X}\) if for every sequence \( \{L_n\}_{n=1}^\infty \) of positive linear operators acting on \(\mathfrak {X}\) satisfying \(\sup _{n \in \mathbb {N}}\Vert L_n\Vert <\infty \) and

then, it holds that

Namely, Korovkin-type results aim to find out which functions form Korovkin sets of a given function space. Note that, in other words, Korovkin’s first theorem asserts that the set \( \{e_0, e_1, e_2\} \subset C([0, 1]) \) is a Korovkin set of *C*([0, 1]) . We next give several examples of Korovkin sets of \( C_\infty ([0, \infty )) \).

### Proposition B.3

(cf. [1, Corollary 6.7]) Suppose that \( 0<\lambda _1<\lambda _2<\lambda _3 \). Then, \(\{f_{\lambda _1}, f_{\lambda _2}, f_{\lambda _3}\}\) is a Korovkin set of \( C_\infty ([0, \infty )) \), where \(f_{\lambda _k}(x):=\exp (-\lambda _k x)\) for \(k=1, 2, 3\).

Moreover, we obtain an easy way to find out Korovkin sets of \( C_\infty ^w([0, \infty )) \) when the weight function *w* is supposed to be bounded.

### Proposition B.4

(cf. [1, Proposition 6.16]) Let \( w \in C_b([0, \infty )) \). Then, every Korovkin set of \( C_\infty ([0, \infty )) \) is also a Korovkin set of \(C_\infty ^w([0, \infty ))\). In particular, the set \(\{f_{\lambda _1}, f_{\lambda _2}, f_{\lambda _3}\}\) as in Proposition B.3 is a Korovkin set of \(C_\infty ^w(E[0, \infty )\).

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Akahori, J., Namba, R. & Semba, S. Limit Theorems for Iterates of the Szász–Mirakyan Operator in Probabilistic View.
*J Theor Probab* **36**, 1321–1338 (2023). https://doi.org/10.1007/s10959-022-01199-5

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DOI: https://doi.org/10.1007/s10959-022-01199-5