Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
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.


References [Enhancements On Off] (What's this?)

  • 1. F. Altomare and M. Campiti, Korovkin type Approximation Theory and its Applications, Walter de Gryter Publ. Berlin, 1994. MR 95g:41001
  • 2. 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.
  • 3. 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 49:8099
  • 4. Y. S. Chow and H. Teicher, Probability theory, independence, interchangeability, martingales, Second edition, Springer Verlag, 1988. MR 89e:60001
  • 5. K. L. Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York, 1974. MR 49:11579
  • 6. P. Diaconis, Weak and strong averages in probability theory and the theory of numbers, Ph. D. thesis, Dept of Statistics, Harvard University, May 1974.
  • 7. P. Embrechts, and M. Maejima, The central limit theorem for summability methods of i.i.d. random variables, Z. Wahrsc. Verw. Gebiete. 68 (1984), no. 2, 191-204. MR 86f:60038
  • 8. W. Feller, An introduction to probability theory and it's applications, Vol. II, Second edition, John Wiley and Sons, New York, 1966. MR 35:1048
  • 9. B. R Gelbaum, Some theorems in probability theory, Pacific J. Math. 118 (1985), no 2, 383-391. MR 86i:60042
  • 10. H. U. Gerber, The discounted central limit theorem and its Berry-Esseen analogs, Ann. Math. Statist. 42 (1971), 389-392. MR 43:1255
  • 11. G. H. Hardy, Divergent series, Clarendon Press, Oxford, 1949. MR 11:25a
  • 12. Y. Kasahara, and M. Maejima, Functional limit theorems for weighted sums of i.i.d. random variables, Probab. Theory Relat. Fields. 72 (1986), no. 2, 161-183. MR 88b:60086
  • 13. T. L. Lai, Summability methods for independent, identically distributed random variables, Proc. Amer. Math. Soc. 45 (1974), 253-261. MR 50:8665
  • 14. E. Omey, A limit theorem for discounted sums, Z. Wahrsch. Verw. Gebiete. 68 (1984), no. 1, 49-51. MR 86c:60034
  • 15. R. L. Powell, and S. M. Shah, Summability theory and applications, Van Nostrand Reinhold, London, 1972.
  • 16. W. E. Pruitt, Summability of independent random variables, J. Math. Mech., 15 (1966), no. 5, 769-776. MR 33:3338

Similar Articles

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

DOI: https://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

American Mathematical Society