Let $(X_n, \mathscr{F}_n, n \geqq 1)$ be a martingale difference sequence with $E(X_n^2 \mid \mathscr{F}_{n-1}) = 1$ a.s. This paper presents iterated logarithm results involving $\lim \sup_{n\rightarrow\infty} \sum^n_{m=1} f(m/n)X_m/(2n \log \log n)^{\frac{1}{2}}$, where $f$ is a continuous function on [0, 1]. For example, it is shown that the above limit superior equals the $L_2$-norm of $f$ if the $X_n$'s are uniformly bounded and $f$ is a power series with radius in excess of one. These results generalize (and correct the proof of) a previous theorem due to the author. A generalization of the strong law of large numbers is also established.
Publié le : 1975-04-14
Classification:
Law of the iterated logarithm,
martingales,
independent random variables,
function of bounded variation,
strong law of large numbers,
60F15,
60G45,
26A45,
60G50
@article{1176996401,
author = {Tomkins, R. J.},
title = {Iterated Logarithm Results for Weighted Averages of Martingale Difference Sequences},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 307-314},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996401}
}
Tomkins, R. J. Iterated Logarithm Results for Weighted Averages of Martingale Difference Sequences. Ann. Probab., Tome 3 (1975) no. 6, pp. 307-314. http://gdmltest.u-ga.fr/item/1176996401/