Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that $$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$ where $\mu$ is close (or going) to zero. We also assume that the moment generating function and its derivatives $M_{X_i}, M_{X_i}', M_{X_i}'', M_{X_i}'''$ are all bounded in $(-\epsilon, \epsilon)$ by a constant $C$, where $\epsilon, C$ are independent of $\mu$ (which is going to zero).

Fix a constant $a\geq 1$, are there any estimates in the literature for the probability of the random walk always stays below $a$, i.e. $$\mathbb{P}\big\{{\max_{n\geq 0} S_n \leq a}\big\}?$$ (I believe the upper bound should be $Ca\mu$.)

For the special case where we replace $a$ by $0$, then $$\mathbb{P}\big\{{\max_{n\geq 0} S_n \leq 0}\big\} \leq C\mu$$ which essentially follows from Sparre-Andersen theorem together with Berry-Esseen bound.