Relationships between the growth of a sequence $N_k$ and conditions on the tail of the distribution of a sequence $X_l$ of i.i.d. mean zero random variables are given that are necessary and sufficient for $$\sum^\infty_{k=1} P\big\{\big|\frac{1}{N_k} \sum^{N_k}_{l=1} X_l\big| > \varepsilon\big\} < \infty.$$ The results are significant for distributions satisfying $E(|X_l|) < \infty$ but $E(|X_l|^\beta) = \infty$ for some $\beta > 1$. Necessary and sufficient conditions for the finiteness of sums of the form $$\sum^\infty_{n=1} \gamma (n)P\big\{\big|\frac{1}{n} \sum^n_{l=1} X_l\big| > \varepsilon\big\}$$ are obtained as a corollary.
Publié le : 1980-02-14
Classification:
Law of large numbers,
triangular array,
complete convergence,
error estimates,
60F10,
60F05,
60F15
@article{1176994835,
author = {Asmussen, Soren and Kurtz, Thomas G.},
title = {Necessary and Sufficient Conditions for Complete Convergence in the Law of Large Numbers},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 176-182},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994835}
}
Asmussen, Soren; Kurtz, Thomas G. Necessary and Sufficient Conditions for Complete Convergence in the Law of Large Numbers. Ann. Probab., Tome 8 (1980) no. 6, pp. 176-182. http://gdmltest.u-ga.fr/item/1176994835/