On BerryEsseen bounds of summability transforms
Authors:
J. A. Fridy, R. A. Goonatilake and M. K. Khan
Journal:
Proc. Amer. Math. Soc. 132 (2004), 273282
MSC (2000):
Primary 60F05; Secondary 41A36, 40C05
Published electronically:
April 24, 2003
MathSciNet review:
2021271
Fulltext PDF Free Access
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 (BerryEsseen 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.
 1.
Francesco
Altomare and Michele
Campiti, Korovkintype 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
(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 0343357
(49 #8099)
 4.
Yuan
Shih Chow and Henry
Teicher, Probability theory, 2nd ed., Springer Texts in
Statistics, SpringerVerlag, New York, 1988. Independence,
interchangeability, martingales. MR 953964
(89e:60001)
 5.
Kai
Lai Chung, A course in probability theory, 2nd ed., Academic
Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1974. Probability and Mathematical Statistics, Vol. 21. MR 0346858
(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.
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
(86f:60038), http://dx.doi.org/10.1007/BF00531777
 8.
William
Feller, An introduction to probability theory and its applications.
Vol. II, John Wiley & Sons, Inc., New YorkLondonSydney, 1966. MR 0210154
(35 #1048)
 9.
Bernard
R. Gelbaum, Some theorems in probability theory, Pacific J.
Math. 118 (1985), no. 2, 383–391. MR 789178
(86i:60042)
 10.
Hans
U. Gerber, The discounted central limit theorem and its
BerryEsséen analogue, Ann. Math. Statist. 42
(1971), 389–392. MR 0275501
(43 #1255)
 11.
G.
H. Hardy, Divergent Series, Oxford, at the Clarendon Press,
1949. MR
0030620 (11,25a)
 12.
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
(88b:60086), http://dx.doi.org/10.1007/BF00699101
 13.
Tze
Leung Lai, Summability methods for independent
identically distributed random variables, Proc.
Amer. Math. Soc. 45
(1974), 253–261. MR 0356194
(50 #8665), http://dx.doi.org/10.1090/S00029939197403561944
 14.
E.
Omey, A limit theorem for discounted sums, Z. Wahrsch. Verw.
Gebiete 68 (1984), no. 1, 49–51. MR 767443
(86c:60034), http://dx.doi.org/10.1007/BF00535172
 15.
R. L. Powell, and S. M. Shah, Summability theory and applications, Van Nostrand Reinhold, London, 1972.
 16.
William
E. Pruitt, Summability of independent random variables, J.
Math. Mech. 15 (1966), 769–776. MR 0195135
(33 #3338)
 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, 207220. 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, 191204. 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, 383391. MR 86i:60042
 10.
 H. U. Gerber, The discounted central limit theorem and its BerryEsseen analogs, Ann. Math. Statist. 42 (1971), 389392. 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, 161183. MR 88b:60086
 13.
 T. L. Lai, Summability methods for independent, identically distributed random variables, Proc. Amer. Math. Soc. 45 (1974), 253261. MR 50:8665
 14.
 E. Omey, A limit theorem for discounted sums, Z. Wahrsch. Verw. Gebiete. 68 (1984), no. 1, 4951. 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, 769776. 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:
http://dx.doi.org/10.1090/S0002993903069879
PII:
S 00029939(03)069879
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
