Small deviations in the sample function space

Let $\xi_1\xi_2,\dots$ be a sequence of independent identically distributed random variables and let
$$ \mathbf P(\xi_1+\dots+\xi_n<xB(n))\Rightarrow F_\alpha(x) $$
where $F_\alpha(x)$ is a strong stable distribution function. Asymptotic properties (in the region of small deviations) of the logarithmic probability for sample paths of a random walk generated by sums of $\xi_n$ to belong to a given set in $D(0,1)$ are under investigation.