## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 1 (1979), 85-95.

### Laws of Large Numbers for $D\lbrack0, 1\rbrack$

Peter Z. Daffer and Robert L. Taylor

#### Abstract

Laws of large numbers are obtained for random variables taking their values in $D\lbrack 0, 1\rbrack$ where $D\lbrack 0, 1\rbrack$ is equipped with the Skorokhod topology. The strong law of large numbers is obtained for independent, convex tight random elements $\{X_n\}$ satisfying $\sup_nE\|X_n\|^r_\infty < \infty$ for some $r > 1$ where $\|X\|_\infty = \sup_{0 \leqslant t \leqslant 1}|x(t)|$. A strong law of large numbers is also obtained for almost surely monotone random elements in $D\lbrack 0, 1\rbrack$ for which the hypothesis of convex tightness is not needed. A discussion of the condition of convex tightness is also included.

#### Article information

**Source**

Ann. Probab., Volume 7, Number 1 (1979), 85-95.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995150

**Digital Object Identifier**

doi:10.1214/aop/1176995150

**Mathematical Reviews number (MathSciNet)**

MR515815

**Zentralblatt MATH identifier**

0402.60010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B05: Probability measures on topological spaces

Secondary: 60F15: Strong theorems 60G99: None of the above, but in this section

**Keywords**

Skorokhod laws of large numbers convex tightness compact convergence almost surely and in probability

#### Citation

Daffer, Peter Z.; Taylor, Robert L. Laws of Large Numbers for $D\lbrack0, 1\rbrack$. Ann. Probab. 7 (1979), no. 1, 85--95. doi:10.1214/aop/1176995150. https://projecteuclid.org/euclid.aop/1176995150