Some results giving rates of convergence in the law of large numbers for weighted sums of independent random variables
Authors:
W. E. Franck and D. L. Hanson
Journal:
Trans. Amer. Math. Soc. 124 (1966), 347359
MSC:
Primary 60.30
MathSciNet review:
0199877
Fulltext PDF Free Access
References 
Similar Articles 
Additional Information
 [1]
D.
L. Hanson and L.
H. Koopmans, On the convergence rate of the law of large numbers
for linear combinations of independent random variables, Ann. Math.
Statist. 36 (1965), 559–564. MR 0182043
(31 #6267)
 [2]
Herman
Chernoff, A measure of asymptotic efficiency for tests of a
hypothesis based on the sum of observations, Ann. Math. Statistics
23 (1952), 493–507. MR 0057518
(15,241c)
 [3]
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)
 [4]
William
E. Pruitt, Summability of independent random variables, J.
Math. Mech. 15 (1966), 769–776. MR 0195135
(33 #3338)
 [5]
D.
L. Hanson and L.
H. Koopmans, Convergence rates for the law of large numbers for
linear combinations of exchangeable and *mixing stochastic processes,
Ann. Math. Statist. 36 (1965), 1840–1852. MR 0185643
(32 #3105)
 [6]
R.
V. Chacon, Ordinary means imply recurrent
means, Bull. Amer. Math. Soc. 70 (1964), 796–797. MR 0168720
(29 #5977), http://dx.doi.org/10.1090/S000299041964112404
 [7]
Melvin
L. Katz, The probability in the tail of a distribution, Ann.
Math. Statist. 34 (1963), 312–318. MR 0144369
(26 #1914)
 [8]
Leonard
E. Baum and Melvin
Katz, Convergence rates in the law of large
numbers, Bull. Amer. Math. Soc. 69 (1963), 771–772. MR 0156373
(27 #6296), http://dx.doi.org/10.1090/S000299041963110277
 [9]
, Convergence rates in the law of large numbers. II, Tech. Rep. No. 75, Univ. of New Mexico Albuquerque, Dept. of Math., 1964.
 [10]
P.
Erdös, On a theorem of Hsu and Robbins, Ann. Math.
Statistics 20 (1949), 286–291. MR 0030714
(11,40f)
 [11]
Michel
Loève, Probability theory, Third edition, D. Van
Nostrand Co., Inc., Princeton, N.J.Toronto, Ont.London, 1963. MR 0203748
(34 #3596)
 [12]
David
Blackwell and J.
L. Hodges Jr., The probability in the extreme tail of a
convolution, Ann. Math. Statist. 30 (1959),
1113–1120. MR 0112197
(22 #3052)
 [1]
 D. L. Hanson and L. H. Koopmans, On the convergence rate of the law of large numbers for linear combinations of independent random variables, Ann. Math. Statist. 36 (1965), 559564. MR 0182043 (31:6267)
 [2]
 Hermann Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statist. 23 (1952), 493507. MR 0057518 (15:241c)
 [3]
 B. Jamison, S. Oreye and W. Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrscheinlichkeitstheorie 4 (1965), 4044. MR 0182044 (31:6268)
 [4]
 William E. Pruitt, Summability of independent random variables, J. Math. Mech. (to appear). MR 0195135 (33:3338)
 [5]
 D. L. Hanson and L. H. Koopmans, Convergence rates for the law of large numbers for linear combinations of exchangeable and mixing stochastic processes, Ann. Math. Statist. 36 (1965), 18401852. MR 0185643 (32:3105)
 [6]
 R. V. Chacon, Ordinary means imply recurrent means, Bull. Amer. Math. Soc. 70 (1964), 796797. MR 0168720 (29:5977)
 [7]
 Melvin L. Katz, The probability in the tail of a distribution, Ann. Math. Statist. 34 (1963), 312318. MR 0144369 (26:1914)
 [8]
 L. E. Baum and Melvin Katz, Convergence rates in the law of large numbers, Bull. Amer. Math. Soc. 69 (1963), 771772. MR 0156373 (27:6296)
 [9]
 , Convergence rates in the law of large numbers. II, Tech. Rep. No. 75, Univ. of New Mexico Albuquerque, Dept. of Math., 1964.
 [10]
 P. Erdös, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), 286291. MR 0030714 (11:40f)
 [11]
 M. Loève, Probability theory, 3rd ed., Van Nostrand, Princeton, N. J., 1963. MR 0203748 (34:3596)
 [12]
 David Blackewel and J. L. Hodges, Jr., The probability in the extreme tail of a convolution, Ann. Math. Statist. 30 (1959), 11131120. MR 0112197 (22:3052)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947196601998771
PII:
S 00029947(1966)01998771
Article copyright:
© Copyright 1966
American Mathematical Society
