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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.

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    $\begingroup$ If $(Z_t)_{t\ge 0}$ is a Brownian motion with drift $-1$ and variance parameter 1, then defining $Z^\mu_t=Z_{\mu^2t}/\mu$, $Z^\mu$ is a Brownian motion with drift $-\mu$ and variance parameter 1. If the random variable $t^*$ is the maximizer of the original BM, them the random variable $t^*/\mu^2$ is the maximizer of $Z^\mu$. $\endgroup$ – Anthony Quas Jan 4 '15 at 4:04
  • $\begingroup$ Thanks! Then how to generalize this result to general random walks? $\endgroup$ – John Wong Jan 4 '15 at 5:00
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    $\begingroup$ One way is to use coupling of the random walk to Brownian motion, e.g., Skorohod embedding or the KMT construction. $\endgroup$ – ofer zeitouni Jan 4 '15 at 6:26
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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).

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