Transient phenomena for random walks in the absence of the expected value of jumps

Let $\xi,\xi_1,\xi_2,\dots$ be independent identically distributed random variables, and
$$ S_n:=\sum_{j=1}^n\xi_j,\qquad\overline S:=\sup_{n\ge0}S_n. $$
If $\mathbf E\xi=-a<0$ then we call transient those phenomena that happen to the distribution $\overline S$ as $a\to0$ and $\overline S$ tends to infinity in probability. We consider the case when $\mathbf E\xi$ fails to exist and study transient phenomena as $a\to0$ for the following two random walk models:
1. The first model assumes that $\xi_j$ can be represented as $\xi_j=\zeta_j+a\eta_j$, where $\zeta_1,\zeta_2,\dots$ and $\eta_1,\eta_2,\dots$ are two independent sequences of independent random variables, identically distributed in each sequence, such that $\sup_{n\ge0}\sum_{j=1}^n\zeta_j=\infty$, $\sup_{n\ge0}\sum_{j=1}^n\eta_j=\infty$, and $\overline S<\infty$ almost surely.
2. In the second model we consider a triangular array scheme with parameter $a$ and assume that the right tail distribution $\mathbf P(\xi_j\ge t)\sim V(t)$ as $t\to\infty$ depends weakly on $a$, while the left tail distribution is $\mathbf P(\xi_j<-t)=W(t/a)$, where $V$ and $W$ are regularly varying functions and $\overline S<\infty$ almost surely for every fixed $a>0$.
We obtain some results for identically and differently distributed $\xi_j$.