Suppose we have some discrete random walk defined $X_0 = 0$ and $X_i = X_{i-1} + Y_i$, where the $Y_i$ are iid random variables with mean $0$ and variance $1$. Using the central limit theorem, one can approximate $\mathbb{P}[X_i > f(i)]$ for some function $f$. Using linearity of expectations, one can then find the expected number of times that $X_i > f(i)$ in some interval.
My question is whether there is a way to prove anything about the distribution of the number of times that $X_i > f(i)$ holds. I would also be interested if there was a way to do this assuming something about $Y_i$, for example if $Y_i$ is $N(0,1)$.