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
- 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, DOI 10.1515/9783110884586
- 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, DOI 10.1007/978-1-4684-0504-0
- Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. 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 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 10.1214/aoms/1177693529
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- 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 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 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 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
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