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.

  • 2
    $\begingroup$ 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
  • 2
    $\begingroup$ 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

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).

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.