On the rate of convergence of one linear positive operator for functions of \(p\)-bounded variation (Q1917707)

It is considered a sequence of linear positive operators of the form \[ A_n(f,x)=\int^\infty_0 f(a_nu)dF_{n,x}(u), \] where \(f\) is a well-defined measurable function on \([0,\infty)\), \(x\in [0,\infty)\), \(\{a_n\}\) is a sequence of positive numbers and \(F_{n,x}(u)\) is the distribution function of \(S_n:=x_{1,n}+\cdots+ x_{n,n}\), where for any \(n\), \(x_{i,n}\) \((1\leq i\leq n)\) are identically distributed independent random variables. If \(f\) has a \(p\)-bounded variation on any finite interval \([0,a]\) and there are the constants \(A>0\) and \(M>0\) such that \(f(x)=O(e^{Ax})\), \((x\to\infty)\) and \(E(e^{2Aa_nS_n})\leq M\), \((n\in\mathbb{N})\), then one obtains an estimate of \[ |A_n(f,x)-(1/2)(f(x+)+f(x-))|, \] using the \(p\)-variation of the function \(g_x\), \(g_x(t)=f(t)-f(x-)\), \((0\leq t<x)\); \(g_x(x)=x\); \(g_x(t)=f(t)-f(x+)\), \((x<t\leq 1)\). Also it is given a variant of the estimation for functions that admit right-hand and left-hand derivatives at \(x\). There are some slight misprints that can be easy corrected. Among other misprints we remark that in Theorem 1.2 the function \(f\) is not explicitely given in the statement of the theorem; it is defined in Lemma 2.2. The function \(g_x\) is defined in connection with the function \(f\). Also the second term in the right member of the inequality given in this theorem seems to be equal to \((1/2)(\xi_n(x))^{1/2}|f(x+)-f(x-)|\) instead of \((Q_n(x)/(2\sqrt n))|f(x+)-f(x-)|\).