## On Berry-Esseen bounds of summability transforms

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- by J. A. Fridy, R. A. Goonatilake and M. K. Khan PDF
- Proc. Amer. Math. Soc.
**132**(2004), 273-282 Request permission

## 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.## References

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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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 273-282 - MSC (2000): Primary 60F05; Secondary 41A36, 40C05
- DOI: https://doi.org/10.1090/S0002-9939-03-06987-9
- MathSciNet review: 2021271