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

DOI:
https://doi.org/10.1090/S0002-9939-03-06987-9

Published electronically:
April 24, 2003

MathSciNet review:
2021271

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Francesco Altomare and Michele Campiti,
*Korovkin-type approximation theory and its applications*, De Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter & Co., Berlin, 1994. Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. MR**1292247** - N. L. Bowers; H. U. Gerber; J. C. Hickman; D. A. Jones, and C. J. Nesbitt,
*Actuarial Mathematics*, Second edition, The Society of Actuaries, Schaumburg, Illinois, 1999. - Y. S. Chow,
*Delayed sums and Borel summability of independent, identically distributed random variables*, Bull. Inst. Math. Acad. Sinica**1**(1973), no. 2, 207–220. MR**343357** - Yuan Shih Chow and Henry Teicher,
*Probability theory*, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1988. Independence, interchangeability, martingales. MR**953964** - Kai Lai Chung,
*A course in probability theory*, 2nd ed., Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 21. MR**0346858** - P. Diaconis,
*Weak and strong averages in probability theory and the theory of numbers*, Ph. D. thesis, Dept of Statistics, Harvard University, May 1974. - Paul Embrechts and Makoto Maejima,
*The central limit theorem for summability methods of i.i.d. random variables*, Z. Wahrsch. Verw. Gebiete**68**(1984), no. 2, 191–204. MR**767800**, DOI https://doi.org/10.1007/BF00531777 - William Feller,
*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154** - Bernard R. Gelbaum,
*Some theorems in probability theory*, Pacific J. Math.**118**(1985), no. 2, 383–391. MR**789178** - Hans U. Gerber,
*The discounted central limit theorem and its Berry-Esséen analogue*, Ann. Math. Statist.**42**(1971), 389–392. MR**275501**, DOI https://doi.org/10.1214/aoms/1177693529 - G. H. Hardy,
*Divergent series*, Clarendon Press, Oxford, 1949. - Yuji Kasahara and Makoto Maejima,
*Functional limit theorems for weighted sums of i.i.d. random variables*, Probab. Theory Relat. Fields**72**(1986), no. 2, 161–183. MR**836273**, DOI https://doi.org/10.1007/BF00699101 - Tze Leung Lai,
*Summability methods for independent identically distributed random variables*, Proc. Amer. Math. Soc.**45**(1974), 253–261. MR**356194**, DOI https://doi.org/10.1090/S0002-9939-1974-0356194-4 - E. Omey,
*A limit theorem for discounted sums*, Z. Wahrsch. Verw. Gebiete**68**(1984), no. 1, 49–51. MR**767443**, DOI https://doi.org/10.1007/BF00535172 - R. L. Powell, and S. M. Shah,
*Summability theory and applications*, Van Nostrand Reinhold, London, 1972. - William E. Pruitt,
*Summability of independent random variables*, J. Math. Mech.**15**(1966), 769–776. MR**0195135**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
60F05,
41A36,
40C05

Retrieve articles in all journals with MSC (2000): 60F05, 41A36, 40C05

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

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