Functional law of the iterated logarithm for truncated sums

We obtain the functional law of the iterated logarithm (the FLIL) for truncated sums $S_n=\sum\limits_{j=1}^n\,X_j\,I\{X^2_j\le b_n\}$ of independent symmetric random variables $X_j$, $1\le j\le n$, $b_n\le\infty$. Considering the random normalization by
$$ T^{1/2}_n=\Bigl(\sum_{j=1}^n\,X^2_j\,I\{X^2_j\le b_n\}\Bigr)^{1/2} $$
we get the upper estimate in the FLIL using only the condition that $T_n\to\infty$ a.s. These results are useful for studing trimmed sums.