The Annals of Probability

On the Rate of Convergence in the Weak Law of Large Numbers

Peter Hall

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Abstract

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.

Article information

Source
Ann. Probab., Volume 10, Number 2 (1982), 374-381.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993863

Digital Object Identifier
doi:10.1214/aop/1176993863

Mathematical Reviews number (MathSciNet)
MR647510

Zentralblatt MATH identifier
0484.60017

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks 60F25: $L^p$-limit theorems

Keywords
Independent and identically distributed random variables rate of convergence weak law of large numbers

Citation

Hall, Peter. On the Rate of Convergence in the Weak Law of Large Numbers. Ann. Probab. 10 (1982), no. 2, 374--381. doi:10.1214/aop/1176993863. https://projecteuclid.org/euclid.aop/1176993863


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