Motivated by the problem of establishing laws of the iterated logarithm for least squares estimates in regression models and for partial sums of linear processes, we prove a general $\log \log$ law for weighted sums of the form $\sum^\infty_{i=-\infty} a_{ni}\varepsilon_i$, where the $\varepsilon_i$ are independent random variables with zero means and a common variance $\sigma^2$, and $\{a_{ni}: n \geq 1, -\infty < i < \infty\}$ is a double array of constants such that $\sum^\infty_{i=-\infty} a^2_{ni} < \infty$ for every $n$. Besides applying the general theorem to least squares estimates and linear processes, we also use it to improve earlier results in the literature concerning weighted sums of the form $\sum^n_{i=1} f(i/n)\varepsilon_i$.
Publié le : 1982-05-14
Classification:
Law of the iterated logarithm,
double arrays,
least squares estimates,
linear processes,
exponential bounds,
moment inequalities,
60F15,
60G35,
62J05,
62M10
@article{1176993860,
author = {Lai, Tze Leung and Wei, Ching Zong},
title = {A Law of the Iterated Logarithm for Double Arrays of Independent Random Variables with Applications to Regression and Time Series Models},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 320-335},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993860}
}
Lai, Tze Leung; Wei, Ching Zong. A Law of the Iterated Logarithm for Double Arrays of Independent Random Variables with Applications to Regression and Time Series Models. Ann. Probab., Tome 10 (1982) no. 4, pp. 320-335. http://gdmltest.u-ga.fr/item/1176993860/