## The Annals of Probability

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

Peter Hall

#### 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

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