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February, 1979 Laws of Large Numbers for $D\lbrack0, 1\rbrack$
Peter Z. Daffer, Robert L. Taylor
Ann. Probab. 7(1): 85-95 (February, 1979). DOI: 10.1214/aop/1176995150


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.


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Peter Z. Daffer. Robert L. Taylor. "Laws of Large Numbers for $D\lbrack0, 1\rbrack$." Ann. Probab. 7 (1) 85 - 95, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0402.60010
MathSciNet: MR515815
Digital Object Identifier: 10.1214/aop/1176995150

Primary: 60B05
Secondary: 60F15 , 60G99

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

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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