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.
Best Answer
Let $$p(a):=P\big(\max_{n\ge0} S_n\le a\big).$$ Assume that $c_3:=E|X_1-EX_1|^3<\infty$.
By the improvement by Sakhanenko of Lemma 8 by S. Nagaev (the improvement consisting in removing an extra factor $c_3$) and trivial time-rescaling, $$p(a)-p(0)\le C\mu(a+c_3)$$ for $\mu\ge0$ and $a\ge0$; everywhere here, $C$ denotes various universal positive real constants. Also, you know that $p(0)\le C\mu$. So, for all $a\ge c_3$ $$p(a)\le C\mu a.\tag{1}$$
Concerning a lower bound on $p(a)$, Corollary 1 to Theorem 16 of $\S$23 of the book by Borovkov implies $$p(a)\ge1-e^{-Qa} \tag{2}$$ for $a\ge0$, where $Q:=\sup\{t\colon M(t)\le1\}$, $M(t):=Ee^{tX}$, and $X:=X_1$.
If e.g. $|X|\le b$ almost surely for some real $b>0$, then for all $t\in[0,Q]$ we have $M''(t)=EX^2e^{tX}\le b^2 M(t)\le b^2$, by the convexity of $M$. So, $1=M(Q)\le M(0)+M'(0)Q+b^2Q^2/2=1-\mu Q+b^2Q^2/2$, whence $Q\ge c\mu$, where $c:=\frac2{b^2}$. If now $Qa$ is large, then (2) implies $p(a)\approx1$, so that the lower bound in (2) is as good as it can be. If, finally, $Qa$ is not large, then the lower bound in (2) is $\asymp Qa\ge c\mu a$, which matches (1).