Asymptotic behavior of operators of probabilistic type in \(L_ p\) spaces (Q1343424)

Let \(X_ 1,X_ 2,\dots\) be independent and identically distributed random variables with mean \(x\) and variance \(\delta^ 2(x)\) depending on the parameter \(x \in J \subset \mathbb{R}\). Let \(S_ n = X_ 1 + \cdots + X_ n\). Then an operator of probabilistic type is defined for bounded continuous functions \(f\) on \(\mathbb{R}\) by \[ P_ n(f,x) = Ef(S_ n/n) = \int f(t/n) dF_{n,x} (t), \] where \(E\) denotes expectation and \(dF_{n,x}\) is the distribution function of \(S_ n\). Several known operators (e.g. Bernstein's) are of this type. The authors study the limits of \(\sqrt n(P_ n (f,x) - f(x))\) and \(n(P_ n (f,x) - f(x))\) pointwise and in the \(L^ p\)-norm. The work follows closely \textit{R. A. Khan}, J. Approximation Theory 45, 339-349 (1985; Zbl 0576.41010). Reviewer's remark: Theorem 1 as stated is wrong. The limits in (1) must be 0. This follows either from the proof (the standard normal variable \(Z\) has zero expectation) or from theorem 2.