Consider a simple symmetric random walk on $\mathbb{Z}$ starting from $0$, $S_n$. Let $I_n := \inf\{S_0, S_1, S_2, \ldots S_n\}$. Is an explicit formula for $E[I_n]$ known?

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    $\begingroup$ Yes. Reflection principle. $E(I_n)\approx -E(|S_n|)$. You can have an precise formula as well. $\endgroup$ – A.S. Mar 17 '16 at 12:39
  • $\begingroup$ It's not entirely fresh in my mind, but iirc this last quantity is asymptotically $-c\sqrt{n}$ for some absolute constant $c>0$ (Update: see (39) in mathworld.wolfram.com/RandomWalk1-Dimensional.html, and (34-35) for non-asymptotics) $\endgroup$ – Clement C. Mar 17 '16 at 12:51
  • $\begingroup$ Thanks, it is known that it grows asymptotically as $\sqrt{n}$, but I am interested in the explicit formula, in case there is an easy one. $\endgroup$ – QuantumLogarithm Mar 17 '16 at 12:56
  • $\begingroup$ Then check the keywords reflection principle mentioned in the first comment. $\endgroup$ – Did Mar 19 '16 at 9:21

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