On Berry-Esseen bounds of summability transforms

Abstract

Abstract. Let Y n,k , k = 0, 1, 2, · · · , n ≥ 1, be a collection of random variables, where for each n, Y n,k , k = 0, 1, 2, · · · , 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 2 -negligibility. An application of these results characterizes the rate of convergence of Schnabl operators while approximating Lipschitz continuous functions.

Adaptive Bernstein–von Mises theorems in Gaussian white noise

“…Let F n,α and Φ denote the cdfs of Z n,α and the standard normal distribution respectively. By the Berry-Esseen theorem for infinite arrays (Theorem 3.2 of [21] with summability matrix p n,k ≡ 1), there exists a universal constant C 0 such that…”

Characterizations of strong and statistical convergences

“…Summability of matrices, as defined in ( ( 7)), commonly arises in approximation theory, probability and statistics contexts. For instance, central limit theorem of triangular arrays [5], [11]; limit theorems concerning random matrices [22]; limit theorems concerning order statistics [27]; and various positive linear approximation operators [15], [17], and [11]. In some of these contexts the four summability methods that play a prominent role are the Euler, logarithmic, Cesáro and Abel summability methods.…”