Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, Teugels), the function $$ l(n) = \exp \left( \sum_{k=1}^\infty \frac{(1-\frac{1}{n})^k} {k} \Big( P(S_k > 0) - \rho \Big) \right) $$ is slowly varying. I am interested in learning more about the behavior of $l(n)$ for $n$ large. In particular, I am asking myself whether the following statement can be made:
$$ l(n) = o(n^\epsilon) \ \ \forall \epsilon > 0.$$
I am thankful for any hints, references and ideas.
Edit (sharing my progress/thoughts): By monotone convergence, $$l(n) \to \exp \left( \sum_{k=1}^\infty \frac{1} {k} \Big( P(S_k > 0) - \rho \Big) \right) $$ as $n \to \infty$. If the right hand side is finite, the claim that $l(n) = o (n^\epsilon)$ for all $\epsilon > 0$ is immediate. So, only the case where the sum on the right hand side diverges, has to be considered. Note that Spitzer's condition is equivalent to $P(S_n > 0 ) \to \rho$ as $n \to \infty$.
