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), 347-359

MSC:
Primary 60.30

DOI:
https://doi.org/10.1090/S0002-9947-1966-0199877-1

MathSciNet review:
0199877

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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]**Hermann Chernoff,*A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations*, Ann. Math. Statist.**23**(1952), 493-507. MR**0057518 (15:241c)****[3]**B. Jamison, S. Oreye and W. Pruitt,*Convergence of weighted averages of independent random variables*, Z. Wahrscheinlichkeitstheorie**4**(1965), 40-44. 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), 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)****[7]**Melvin L. Katz,*The probability in the tail of a distribution*, Ann. Math. Statist.**34**(1963), 312-318. 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), 771-772. 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), 286-291. 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), 1113-1120. MR**0112197 (22:3052)**

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DOI:
https://doi.org/10.1090/S0002-9947-1966-0199877-1

Article copyright:
© Copyright 1966
American Mathematical Society