## The Annals of Probability

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

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

#### 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