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

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

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Correspondence to Ryuya Namba.

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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

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert T_n^{[nt]}f - {\mathsf {T}}_tf\Vert _{\mathcal {U}}=0, \qquad t \ge 0, \, f \in \mathcal {U}. \end{aligned}$$

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

$$\begin{aligned} \Vert \mathcal {A}_n f\Vert _{\mathcal {U}} \le \varphi _n(f), \qquad \Vert \mathcal {A}_n f - \mathcal {A}f\Vert _{\mathcal {U}} \le \psi _n(f), \qquad f \in \mathcal {D}, \end{aligned}$$

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

$$\begin{aligned} \Vert T_n^{[nt]}f - {\mathsf {T}}_tf\Vert _{\mathcal {U}} =\sqrt{\frac{t}{n}}\varphi _n(f)+\frac{1}{n}\varphi _n(f) +\int _0^t \psi _n({\mathsf {T}}_sf) \, \mathrm {d}s, \qquad n \in \mathbb {N}. \end{aligned}$$

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

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert L_ne_i - e_i\Vert _\infty =0, \qquad i=0, 1, 2, \end{aligned}$$

then, it holds that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert L_nf - f\Vert _\infty =0, \qquad f \in C([0, 1]). \end{aligned}$$

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

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert L_nf - f\Vert _{\mathfrak {X}}=0, \qquad f \in \mathcal {K}, \end{aligned}$$

then, it holds that

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert L_nf - f\Vert _{\mathfrak {X}}=0, \qquad f \in \mathfrak {X}. \end{aligned}$$

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).

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