PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 1, Pages 273–282 S 0002-9939(03)06987-9 Article electronically published on April 24, 2003 ON BERRY-ESSEEN BOUNDS OF SUMMABILITY TRANSFORMS J. A. FRIDY, R. A. GOONATILAKE, AND M. K. KHAN (Communicated by Claudia M. Neuhauser) Abstract. Let Y n,k , k =0, 1, 2, ··· , n ≥ 1, be a collection of random vari- ables, 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 trans- form (AY ). We show that when A =[p n,k ] is the classical Ces´aro summa- bility 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 concern- ing ℓ 2 -negligibility. An application of these results characterizes the rate of convergence of Schnabl operators while approximating Lipschitz continuous functions. 1. Introduction Let Y := [Y k,m ] be a matrix of random variables, where (Y k,m , k =0, 1, 2 ··· ) is the m-th column vector consisting of mutually independent random variables with finite variances, m =0, 1, 2, ··· . Let A := [p n,k ] be a summability matrix. Consider S n := ∞  k=0 p n,k (Y n,k − E(Y n,k )) ‖σ n p n ‖ 2 =: ∞  k=0 X n,k , where X n,k := p n,k (Y n,k − E(Y n,k )) ‖σ n p n ‖ 2 , ‖σ n p n ‖ 2 2 := ∞  k=0 p 2 n,k V ar(Y n,k ) < ∞. We will present some results concerning the weak convergence of the sequence S n , n ≥ 1. Since different summability methods have different convergence fields, one expects to see the dependence of rates of convergence of S n on the choice of A. Consequently, a natural question is to ask if there is a summability transform that leads to the fastest rate of convergence. We will show that the classical Ces´ aro transform provides the fastest rate among all regular triangular methods whose row sums equal one. Convergence, in probability and almost sure sense, of such transforms have al- ready been settled by various authors ([3], [4], [13], [16]). In this paper we will show Received by the editors August 3, 2001 and, in revised form, August 22, 2002. 2000 Mathematics Subject Classification. Primary 60F05; Secondary 41A36, 40C05. Key words and phrases. Approximation operators, central limit theorem, convolution methods, Schnabl operators. c 2003 American Mathematical Society 273 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use