Let $\alpha \in (0,2)$, (or for simplicity just $\alpha \in (1,2)$) and let $X_1,X_2,\dots$ be an i.i.d collection of random variables with common distribution $$ p(x,y)= \frac{c_\alpha}{|x-y|^{1+\alpha}}1_{[x\neq 0]} $$ where $c_\alpha = \frac{1}{2\zeta(1+\alpha)}$ is the normalising constant. Now, consider the random walk $\{S_n\}$ given by $S_n = \sum_{i=1}^n X_i$ and $S_0=0$.
For a given $n\ge 1$, consider $\Lambda_n = [-n,n]$, I would like to know whether good bounds for the hitting measure defined as
$$ H_{\Lambda_n}(0,x)= P_0[S_{\tau_n}=x], $$ where $\tau_n = \inf\{n\ge 0, |S_n|\ge n\}$. I have seen such problems under the name of overshoot and undershoot bounds. But I have not found an article dealing with the setting without a drift and with heavy tail random walks. It is easy to get very bad bounds, but then they are far from matching.
From a few simulations I made, it seems like it decays as a power of $(|x|-n)$, but I couldn't figure out what was the exponent for an arbitrary $\alpha$.
Are there good bounds known for $H_{\Lambda_n}(0,x)= P_0[S_{\tau_n}=x]$?
EDIT: As mentioned in the comments bellow, one can use that for $x>n$ $$ (1) \qquad \frac{c_\alpha}{(x+2n)^{1+\alpha}} \le H_{\Lambda_n}(0,x)\le \frac{c_\alpha}{(x-n)^{1+\alpha}} $$ and therefore for $x>cn$ for some $c>1$, we have that $H_{\Lambda_n}(0,x)\asymp n^{-1-\alpha}$. So the question is really about the remaining $x \in [n+1,cn]$. As the bounds given in $(1)$ would lead to upper and lower bounds that that do not match.
