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 | References | Similar Articles | Additional Information

Abstract: Let , , , be a collection of random variables, where for each , , , are independent. Let be a regular summability method. We provide some rates of convergence (Berry-Esseen type bounds) for the weak convergence of summability transform . We show that when 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 -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

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