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.

On the distribution of first hits for the symmetric stable processes, Trans. Am. Math. Soc., 99 (1961), 540–554. Related references: M. Kac,Some remarks on stable processes, Publ. Inst. Stat. Univ. Paris, 6 (1957), 303–306; and: M. Riesz,Intégrales de Riemann–Liouville et potentiels, Acta Sci. Math. Szeged, 9 (1938), 1–42. $\endgroup$9more comments