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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Berry-Esseen bounds of summability transforms


Authors: J. A. Fridy, R. A. Goonatilake and M. K. Khan
Journal: Proc. Amer. Math. Soc. 132 (2004), 273-282
MSC (2000): Primary 60F05; Secondary 41A36, 40C05
Published electronically: April 24, 2003
MathSciNet review: 2021271
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Abstract: 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.


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Additional Information

J. A. Fridy
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: fridy@math.kent.edu

R. A. Goonatilake
Affiliation: Department of Mathematics, Texas A&M International University, Laredo, Texas 78041
Email: harag@tamiu.edu

M. K. Khan
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: kazim@math.kent.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06987-9
PII: S 0002-9939(03)06987-9
Keywords: Approximation operators, central limit theorem, convolution methods, Schnabl operators
Received by editor(s): August 3, 2001
Received by editor(s) in revised form: August 22, 2002
Published electronically: April 24, 2003
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2003 American Mathematical Society