$X_i$ be a simple symmetric random walk. $Y_i= \operatorname{sign}(X_i)$. Define $Z_n= \sum_{i=1}^{n} Yi$. What is an upper bound on $Var(Z_n)$?
I feel the main issue is that $Y_i$ are not independent. Maybe there is some martingale hidden around which I’m not sure.