This is an extended question based on 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.