For a function $f$ consider a random sequence
$a_{n+1}$ can be either $a_n+f(a_n)$ or $a_n-f(a_n)$
Given that the next term in the sequence is subtracting $f(a_n)$ from the previous term 50% of the time and adding it 50% of the time, what function $f$ yields $a_{n+x} > a_n$ for "most" positive values of $x$?