Let $S_n$ be the distance walked at time $n$ in a simple symmetric random walk on $\mathbb{Z}$ (with steps = $\pm1$).

It is well known that $\lim_{n\to\infty}(E(|S_n|))$=$\sqrt{2n/\pi}$.


Can we describe the set $F$ of monotonic "growth" functions with probability 1 of being hit by $|S_n|$ for $n$>1?

Clearly $f(x)=\sqrt{x}$ is in $F$ and I think I could prove that $g(x)=2f(x/2)$ is in $F$ whenever $f$ is. I have no idea beyond that.