A new variant of the functional law of the iterated logarithm
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 502-512
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The full description of the set of limit points of the sequence (3) is given, where $W(t)$
is a $d$-dimensional Brownian motion consisting of $d$ independent Brownian motions and
$\varphi(\,\cdot\,)$ is arbitrary function such that $\varphi(t)\uparrow\infty$ ($t\uparrow\infty$).
We show that with probability one this set coincides with the set $K_{R(\varphi)}$ specified in theorems 1–3.
The sequences of the form (18) are also considered. The result of V. Strassen is a special case
when $\varphi(t)=\sqrt{2\ln\ln t}$. The generalization of Hartman–Wintner's theorem is obtained.
Theorems 4, 5 are valid for all sequences satisfying the almost sure invariance principles
(martingale-differences, sequences with mixing etc.).
@article{TVP_1980_25_3_a4,
author = {A. V. Bulinskiǐ},
title = {A new variant of the functional law of the iterated logarithm},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {502--512},
publisher = {mathdoc},
volume = {25},
number = {3},
year = {1980},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a4/}
}
A. V. Bulinskiǐ. A new variant of the functional law of the iterated logarithm. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 502-512. http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a4/