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.



Random walk always stays below a level $a$
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](https://epubs.siam.org/doi/pdf/10.1137/1119047) of Lemma 8 by [S. Nagaev](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=1854&option_lang=eng) (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 $a\ge c\_3$
$$p(a)\le C\mu a.\tag{1}$$




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Concerning a lower bound on $p(a)$, Corollary 1 to Theorem 16 of $\S$23 of the book by [Borovkov](https://link.springer.com/book/10.1007/978-1-4612-9866-3#:%7E:text=The%20object%20of%20queueing%20theory,on%20the%20direction%20of%20investigation.) 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).


