TITLE: How long for a simple random walk to exceed $\sqrt{T}$?
QUESTION [13 upvotes]: Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$.
What is an expression for the probability distribution of $T$?

REPLY [3 votes]: Building on Yemon, who suggests that the solution is some distribution of a hitting time, if we assume the threshold for the 'hit' is $\sqrt{T}$, and that variance, $u$, of the random variable is directly proportional to the square root of time, then the mean time for that variable to exceed $\sqrt{T}$ may equal $ku\sqrt{T} \Phi(1)/2$, or approximately $ku\sqrt{T} \cdot 0.16$, distributed lognormally.