# The problem is about the expection of the exitpoint distance for the symmetric random walk.

Let $$\nu(x)$$ be a symmetric probability measure with respect to the origin on $$x\in[-1,1]$$ such that $$\nu(\{0\})\neq 1$$.

Consider a random walk started at $$S_0=0$$, denoted $$S_n=X_1+\cdots+X_n$$, where $$X_1,X_2, \cdots$$ are the i.i.d sequences such $$X_i\sim \nu(x)$$. For some $$1\leq L<\infty$$, denote $$\tau=\inf\{n\geq0: S_n>L\}$$.

Let $$\hbar_{\nu,L}=\mathbb{E}(S_\tau)-L$$, in other words, $$\hbar_{\nu,L}$$ is the mean value of exitpoint distance from $$L$$.

$$\textbf{My question is how to derive the explicit formula for}$$ $$\bf{\hbar_{\nu,L}}\textbf{?}$$

Mey be one can start by some simple $$\nu(x)$$ and fix $$L=1$$. Let $$\mu(x)$$ be the probability density function of $$\nu(x)$$, for example,

$$\textbf{(i)}\ \ \$$ $$\mu(x)=1/2,~ x\in[-1,1];$$

$$\textbf{(ii)}\ \ \ \mu(x)=\frac{2}{\pi}\sqrt{1-x^2},~ x\in[-1,1];$$

If possible，could you recommend some relevant papers or books for me？ Anyway, any hints or help would be appreciated. Thank you very much.