I have a sequence $(X_i)_{i\geq 1}$ of i.i.d. random variables taking values in $\mathbb Z$. I know that each $X_i$ has mean $0$ and finite variance $\sigma^2$. Let $S_n=X_1+\cdots+X_n$. Then I can view $(S_n)_{n\geq 1}$ as a random walk on $\mathbb Z$ (with possibly large step sizes). For $B\subseteq\mathbb Z$, let $\tau_B=\inf\{n\geq 1:S_n\in B\}$. Are there any techniques that I can use to obtain upper bounds on the probability that $\tau_B$ lies in some range?
I have two specific settings where I would like to apply such bounds. The first is trying to bound the probability that the random walk veers too far away from the origin in a given amount of time. Let $K$ be a large positive integer (say $K=1000$). Since $\frac{1}{\sqrt{n}}S_n$ should be approximately normally distributed with mean $0$ and variance $\sigma^2$, I should expect that the random walk should not veer more than $K\sigma\sqrt{m}$ from the origin during the first $m$ steps (when $m$ is large). To put this into the above setting, I can let $B=(-\infty,-K\sigma\sqrt{m}]\cup[K\sigma\sqrt{m},\infty)$. Then I want to show that $\mathbb P(\tau_B\leq m)$ is small.
For the second setting, say for concreteness that $B=[3\sigma\sqrt{m},\infty)$ (for $m$ large). I want to show that it is unlikely for $\tau_B$ to lie in some relatively narrow window. For instance, if I take $I=[m-m^{3/4},m+m^{3/4}]$, then I'd like to have some bound saying that $\mathbb P(\tau_B\in I)$ is small.
