On the rate of convergence of some operators on functions of bounded variation (Q1824789)

Let \(\{X_ n,n\geq 1\}\) be a sequence of independent and identically distributed random variables with finite variance such that \(E(X_ 1)=x\in I\subseteq {\mathbb{R}}\), \(Var(X_ 1)=\sigma^ 2(x)>0\), \(S_ n=X_ 1+X_ 2+...+X_ n\). For a function f, Feller defined an approximation operator given by \(L_ n(f,x)=\int f(t/n)dF_{n,x}(t)\), where \(F_{n,x}\) is the distribution function of \(S_ n\) and \(| f|\) is \(F_{n,x}\)-integrable. The rate with which the sequence of operators \(\{L_ n(f,x)\}\) tends to \((f^+,f^ 2)(x)\) pointwise for \(x\in I\) is estimated, where f is of bounded variation on I.