A supermartingale maximal inequality is derived. A maximal inequality is derived for arbitrary random variables $\{S_n, n \geqq 1\}$ (let $S_0 = 0$) satisfying $E\exp\lbrack u(S_{m + n} - S_m) \rbrack \leqq \exp(Knu^2)$ for all real $u$, all integers $m \geqq 0$ and $n \geqq 1$, and some constant $K$. These two maximal inequalities are used to derive upper half laws of the iterated logarithm for supermartingales, multiplicative random variables, and random variables not satisfying particular dependence assumptions.
Publié le : 1973-04-14
Classification:
Law of the iterated logarithm,
maximal inequality,
supermartingale,
martingale,
generalized Gaussian random variables,
multiplicative random variables,
equinormed strongly multiplicative random variables,
60F15,
60G40,
60G45,
60G99
@article{1176996985,
author = {Stout, William F.},
title = {Maximal Inequalities and the Law of the Iterated Logarithm},
journal = {Ann. Probab.},
volume = {1},
number = {5},
year = {1973},
pages = { 322-328},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996985}
}
Stout, William F. Maximal Inequalities and the Law of the Iterated Logarithm. Ann. Probab., Tome 1 (1973) no. 5, pp. 322-328. http://gdmltest.u-ga.fr/item/1176996985/