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 $T$?



How long for a simple random walk to exceed $\sqrt{T}$?
For a Brownian motion, [Novikov](http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=TPRBAU000016000003000449000001&idtype=cvips&gifs=Yes) finds an explicit expression for any real moments (positive and negative) of the random variable $(\tau(a,b,c)+c)$, where
$$
\tau(a,b,c) = \inf(t \geq 0, W(t) \leq -a +b(t+c)^{1/2}) 
$$
with $a \geq 0$, $c \geq 0$, and $bc^{1/2} < a$. [Shepp](https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-38/issue-6/A-First-Passage-Problem-for-the-Wiener-Process/10.1214/aoms/1177698626.full) provides similar results but with W(t) replaced by |W(t)| in the definition, and the range of permissible $a,b,c$ restricted accordingly. Shepp also cites papers by Blackwell and Freedman (1964), Chow, Robbins, and Teicher (1965), and Chow and Teicher (1965), which look like they prove similar but weaker results when the Brownian motion is replaced by a random walk with finite variance. I don't have time to read those references at the moment but I figure these papers should lead you to your answer.


