A lemma concerning real sequences is proved and applied to sequences of random variables $(\mathrm{rv}) X_1, X_2\cdots$ to determine conditions under which $\lim\sup_{n\rightarrow\infty} b_n^{-1} \sum^n_{m=1} f(m/n)X_m < \infty$ a.s. for all $f$ in a particular collection of absolutely continuous functions and for nondecreasing positive real sequences $\{b_n\}$. Theorems in the case $b_n = (2n \log \log n)^\frac{1}{2}$ are proved for generalized Gaussian rv, for equinormed multiplicative systems and for certain martingale difference sequences.
Publié le : 1976-06-14
Classification:
Law of the iterated logarithm,
strong law of large numbers,
multiplicative systems,
generalized Gaussian random variables,
martingale difference sequence,
60F15,
60G10,
60G99
@article{1176996092,
author = {Tomkins, R. J.},
title = {Strong Limit Theorems for Certain Arrays of Random Variables},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 444-452},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996092}
}
Tomkins, R. J. Strong Limit Theorems for Certain Arrays of Random Variables. Ann. Probab., Tome 4 (1976) no. 6, pp. 444-452. http://gdmltest.u-ga.fr/item/1176996092/