On Berry-Esseen bounds of summability transforms

Let $Y_{n,k}$, $k=0, 1,2, \cdots$, $n\geq 1$, be a collection of random variables, where for each $n$, $Y_{n,k}$, $k = 0,1,2,\cdots$, are independent. Let $A=[p_{n,k}]$ be a regular summability method. We provide some rates of convergence (Berry-Esseen type bounds) for the weak convergence of summability transform $(AY)$. We show that when $A=[p_{n,k} ]$ is the classical Cesáro summability method, the rate of convergence of the resulting central limit theorem is best possible among all regular triangular summability methods with rows adding up to one. We further provide some summability results concerning $\ell^2$-negligibility. An application of these results characterizes the rate of convergence of Schnabl operators while approximating Lipschitz continuous functions.