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May, 1982 On the Rate of Convergence in the Weak Law of Large Numbers
Peter Hall
Ann. Probab. 10(2): 374-381 (May, 1982). DOI: 10.1214/aop/1176993863


Let $\{Z_n\}$ be a sequence of random variables converging in probability to zero. If the convergence is also in $L^2$, it is common to measure the rate of convergence by the $L^2$ norm of $Z_n$. However, in many interesting cases the variables $Z_n$ do not have finite variance, and then it seems appropriate to study the truncated $L^2$ norm, $\Delta_n = E\lbrack\min(1, Z^2_n)\rbrack$. We put $\Delta_n$ forward as a global measure of the rate of convergence. The paper concentrates on the case where $Z_n$ is a normalised sum of independent and identically distributed random variables, and we derive very precise descriptions of the rate of convergence in this situation.


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Peter Hall. "On the Rate of Convergence in the Weak Law of Large Numbers." Ann. Probab. 10 (2) 374 - 381, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0484.60017
MathSciNet: MR647510
Digital Object Identifier: 10.1214/aop/1176993863

Primary: 60F05
Secondary: 60F25 , 60G50

Keywords: Independent and identically distributed random variables , rate of convergence , Weak law of large numbers

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • May, 1982
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