On the law of iterated logarithm in Chung's form for functional spaces

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 $(x_1,\dots,x_n)$, $\mathbf P\{x_i<t\}=t\ (0\le t\le 1)$,
$$ K_n=\sup_{0\le t\le 1}|F_n(t)-t|,\qquad \omega_n^2=\int_0^1(F_n(t)-t)^2\,dt. $$
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*}