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.
Publié le : 1979-02-14
Classification:
Skorokhod,
laws of large numbers,
convex tightness,
compact,
convergence almost surely and in probability,
60B05,
60F15,
60G99
@article{1176995150,
author = {Daffer, Peter Z. and Taylor, Robert L.},
title = {Laws of Large Numbers for $D\lbrack0, 1\rbrack$},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 85-95},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995150}
}
Daffer, Peter Z.; Taylor, Robert L. Laws of Large Numbers for $D\lbrack0, 1\rbrack$. Ann. Probab., Tome 7 (1979) no. 6, pp. 85-95. http://gdmltest.u-ga.fr/item/1176995150/