Geodesic
On the law of iterated logarithm in Chung's form for functional spaces
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 399-407 
A. A. Mogul'skiǐ. On the law of iterated logarithm in Chung's form for functional spaces. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 399-407. http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a14/
@article{TVP_1979_24_2_a14,
     author = {A. A. Mogul'skiǐ},
     title = {On the law of iterated logarithm in {Chung's} form for functional spaces},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {399--407},
     year = {1979},
     volume = {24},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a14/}
}
TY  - JOUR
AU  - A. A. Mogul'skiǐ
TI  - On the law of iterated logarithm in Chung's form for functional spaces
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 1979
SP  - 399
EP  - 407
VL  - 24
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a14/
LA  - ru
ID  - TVP_1979_24_2_a14
ER  - 
%0 Journal Article
%A A. A. Mogul'skiǐ
%T On the law of iterated logarithm in Chung's form for functional spaces
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1979
%P 399-407
%V 24
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a14/
%G ru
%F TVP_1979_24_2_a14

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $(X_n)$ be a sequence of independent identically distributed random vectors in a Banach space $(B,\|\cdot\|)$. The paper deals with the following form of the law of iterated logarithm in $B$: with probability 1 $$ \liminf_{n\to\infty}\frac{\|X_1+\dots+X_n\|}{\sqrt n}\Lambda(\ln\ln n)=1. $$ For example, let $F_n(t)$ be the empirical distribution function for a random sample $(x_1,\dots,x_n)$, $\mathbf P\{x_i, $$ K_n=\sup_{0\le t\le 1}|F_n(t)-t|,\qquad \omega_n^2=\int_0^1(F_n(t)-t)^2\,dt. $$ Then with probability 1 \begin{gather*} \liminf_{n\to\infty}K_n\sqrt{n\ln\ln n}=\pi/\sqrt 8, \\ \liminf_{n\to\infty}\omega_n\sqrt{n\ln\ln n}=1/\sqrt 8. \end{gather*}