Let $n \geq 0$, $X_n$ be a random walk, where $X_{n+1} = X_n + 1$ with probability $p$, and $X_{n+1} = X_n - 1$ with probability $1-p$. $X_0 = 0$
Let $l_n, r_n$ be a sequence of integers, where for all $n$, $l_n < 0, r_n > 0$, both take simple closed forms, such as $l_n = \alpha \log{n} + 1$ for some $\alpha$.
The question is: how to obtain a good lower bound on the probability (exact computation would be even better!), given some closed form expression for $l_n$ and $r_n$, that the random walk crosses $r_n$ before it crosses $l_n$, or formally, probability of the event:
\begin{align*} \exists N, X_N \geq r_N, \forall n \leq N, x_n > l_n \end{align*}
Seems like a tough problem, any hints would be appreciated.