The Annals of Probability

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

Peter Hall

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

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