The Annals of Probability

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

Peter Z. Daffer and Robert L. Taylor

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60B05: Probability measures on topological spaces
Secondary: 60F15: Strong theorems 60G99: None of the above, but in this section

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


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.

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