This is an extended question based on
[Large deviations for maximizer of random walk with drift](https://mathoverflow.net/questions/189143/large-deviations-for-maximizer-of-random-walk-with-drift).


Let $$S\_k = X\_1 + \ldots + X\_k,$$ where $X\_i$ are i.i.d. with mean $-\mu < 0$ and unit variance. Assume any nice properties for the tails. Let $k^\star$ be the maximizer of $S\_k$.


Now, we turn to the setting where $\mu \rightarrow 0$. What's the typical size of $k^\star$ in terms of $\mu$? I guess it is $$k^\star = O\_P\left(\frac1{\mu^2}\right).$$


I don't know how to prove it. Any references are highly appreciated.



Maximizer of random walk with very small drift
I looked at this question a while ago, see <http://peter.windridge.org.uk/minRWepsdrift.pdf> . Minimiser is $O\_p(1/\mu^2)$ = Lemma 2.2, using Kolmogorov's max inequality etc. I think that part only uses 2 moments.


Disclaimer: those notes are not the latest version, so might contain minor mistakes and the conditions are definitely not optimal. (I can't find the new version sorry). 


