A Strong Law of Large Numbers for Partial-Sum Processes Indexed by Sets

Bass, Richard F.; Pyke, Ronald

Ann. Probab., Tome 12 (1984) no. 4, p. 268-271 / Harvested from Project Euclid

Résumé

Let $J = \{1, 2, \cdots\}^d$ and let $\{X_j, \mathbf{j} \in J\}$ be iid with finite mean. Let $S(nA)$ be the sum of those $X_j$'s for which $\mathbf{j}/n \in A$. It is proved in this paper that $S(\cdot)$ satisfies a strong law of large numbers that is uniform over $A \in \mathscr{A}$, where $\mathscr{A}$ is a family of subsets of $\lbrack 0, 1\rbrack^d$ satisfying a mild condition.

Publié le : 1984-02-14
Classification: Strong law of large numbers, partial-sum processes, processes indexed by sets, 60F15, 60G99

@article{1176993390,
     author = {Bass, Richard F. and Pyke, Ronald},
     title = {A Strong Law of Large Numbers for Partial-Sum Processes Indexed by Sets},
     journal = {Ann. Probab.},
     volume = {12},
     number = {4},
     year = {1984},
     pages = { 268-271},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176993390}
}

Bass, Richard F.; Pyke, Ronald. A Strong Law of Large Numbers for Partial-Sum Processes Indexed by Sets. Ann. Probab., Tome 12 (1984) no. 4, pp.  268-271. http://gdmltest.u-ga.fr/item/1176993390/