Convergence of weighted averages of random variables revisited
Author:
Nasrollah Etemadi
Journal:
Proc. Amer. Math. Soc. 134 (2006), 27392744
MSC (2000):
Primary 60F15
Published electronically:
April 10, 2006
MathSciNet review:
2213754
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that for a large class of positive weights including the ones that are eventually monotone decreasing and those that are eventually monotone increasing but vary regularly, if the averages of random variables converge in some sense, then their corresponding weighted averages also converge in the same sense. We will also replace the sufficient conditions in the fundamental result of Jamison, Pruitt, and Orey for i.i.d. random variables that make their work more transparent.
 [1]
Xiru
Chen, LiXing
Zhu, and KaiTai
Fang, Almost sure convergence of weighted sums, Statist.
Sinica 6 (1996), no. 2, 499–507. MR 1399318
(97f:60067)
 [2]
S.
Csörgő, K.
Tandori, and V.
Totik, On the strong law of large numbers for pairwise independent
random variables, Acta Math. Hungar. 42 (1983),
no. 34, 319–330. MR 722846
(85e:60034), http://dx.doi.org/10.1007/BF01956779
 [3]
N.
Etemadi, An elementary proof of the strong law of large
numbers, Z. Wahrsch. Verw. Gebiete 55 (1981),
no. 1, 119–122. MR 606010
(82b:60027), http://dx.doi.org/10.1007/BF01013465
 [4]
N.
Etemadi, Stability of sums of weighted nonnegative random
variables, J. Multivariate Anal. 13 (1983),
no. 2, 361–365. MR 705557
(85a:60035), http://dx.doi.org/10.1016/0047259X(83)900325
 [5]
William
Feller, An introduction to probability theory and its applications.
Vol. II., Second edition, John Wiley & Sons, Inc., New
YorkLondonSydney, 1971. MR 0270403
(42 #5292)
 [6]
Seymour
Geisser and Nathan
Mantel, Pairwise independence of jointly dependent variables,
Ann. Math. Statist. 33 (1962), 290–291. MR 0137188
(25 #644)
 [7]
Harber, M. (1986), Testing for pairwise independence, Biometrics 42, 429435.
 [8]
C.
C. Heyde, On almost sure convergence for sums of independent random
variables., Sankhyā Ser. A 30 (1968),
353–358. MR 0279865
(43 #5586)
 [9]
Benton
Jamison, Steven
Orey, and William
Pruitt, Convergence of weighted averages of independent random
variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
4 (1965), 40–44. MR 0182044
(31 #6268)
 [10]
Konrad
Knopp, Theorie und Anwendung der Unendlichen Reihen,
SpringerVerlag, Berlin and Heidelberg, 1947 (German). 4th ed. MR 0028430
(10,446a)
 [11]
HanYing
Liang, Complete convergence for weighted sums of negatively
associated random variables, Statist. Probab. Lett.
48 (2000), no. 4, 317–325. MR 1771494
(2001e:60064), http://dx.doi.org/10.1016/S01677152(00)00002X
 [12]
G.
L. O’Brien, Pairwise independent random variables, Ann.
Probab. 8 (1980), no. 1, 170–175. MR 556424
(81d:60011)
 [13]
Wigderson, A. (1994), The amazing power of pairwise independence. Proceedings of the twentysixth annual ACM symposium on Theory of Computing, Stoc. 645647, Montreal, Canada.
 [1]
 Chen, X., Zhu, L. X., Fang, K.T. (1996), Almost sure convergence of weighted sum. Statistical Sinica, 6, 499507.MR 1399318 (97f:60067)
 [2]
 Csörgö, S., Tandori, K., and Totik, V. (1983), On the strong law of large numbers for pairwise independent random variables, Acta Math. Hung. 42, 319330.MR 0722846 (85e:60034)
 [3]
 Etemadi, N. (1981), An elementary proof of the strong law of large numbers, Z Warrsch. Gebiete 55, 119122. MR 0606010 (82b:60027)
 [4]
 Etemadi, N. (1983), Stability of Sums of Weighted Nonnegative Random Variables, Journal of Multivariate Analysis, Vol. 13, No. 2, 361365. MR 0705557 (85a:60035)
 [5]
 Feller, W. (1971), An Introduction to Probability Theory and Its Applications, Volume II, 2nd Ed., John Wiley & Sons Inc., New York. MR 0270403 (42:5292)
 [6]
 Geisser, S. and Mantel, N. (1962), Pairwise independence of jointly dependent variables, The Annals of Statistics, Vol. 33, No. 1, 290291.MR 0137188 (25:644)
 [7]
 Harber, M. (1986), Testing for pairwise independence, Biometrics 42, 429435.
 [8]
 Heyde, C. C. (1968), On almost sure convergence for sums of independent random variables, Sankhya A 30, 353358. MR 0279865 (43:5586)
 [9]
 Jamison, B., Orey, S., and Pruitt, W. (1965), Convergence of weighted averages of independent random variables, Z. Wahrsch. Gebiete 4, 4044.MR 0182044 (31:6268)
 [10]
 Knopp, K. (1990), Theory and Application of Infinite Series. Dover Publishing Company. MR 0028430 (10:446a)
 [11]
 Liang, H. Y. (2000), Complete convergence for weighted sums of negatively associated random variables, Statistics & Probability Letters, 48, 317325.MR 1771494 (2001e:60064)
 [12]
 O'Brein G. L. (1980), Pairwise independent random variables, The Annals of Probability, Vol. 8, No. 1, 170175. MR 0556424 (81d:60011)
 [13]
 Wigderson, A. (1994), The amazing power of pairwise independence. Proceedings of the twentysixth annual ACM symposium on Theory of Computing, Stoc. 645647, Montreal, Canada.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
60F15
Retrieve articles in all journals
with MSC (2000):
60F15
Additional Information
Nasrollah Etemadi
Affiliation:
Department of Mathematics, Statistics, & Computer Science, University of Illinois at Chicago, 322 Science & Engineering Offices (SEO) m/c 249, 851 S. Morgan Street, Chicago, Illinois 606077045
Email:
Etemadi@uic.edu
DOI:
http://dx.doi.org/10.1090/S0002993906082967
PII:
S 00029939(06)082967
Keywords:
Weighted averages,
limit theorems,
pairwise independence
Received by editor(s):
December 6, 2004
Received by editor(s) in revised form:
March 31, 2005, and April 14, 2005
Published electronically:
April 10, 2006
Additional Notes:
This work was partially supported by the Mahani Mathematical Research Center
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2006
American Mathematical Society
