Local limit theorems for ladder moments

Abstract

Let $S_{0}=0,\{S_{n}\}_{n\geq1}$ be a random walk generated by a sequence of i.i.d. random variables $X_{1},X_{2},...$ and let $\tau^{-}:=\min\left\{ n\geq1:\ S_{n}\leq0\right\}$ and $\tau^{+}:=\min\left\{ n\geq 1:\ S_{n}>0\right\}$. Assuming that the distribution of $X_{1}$ belongs to the domain of attraction of an $\alpha$-stable law$,\alpha\neq1,$ we study the asymptotic behavior of $\mathbb{P}(\tau^{\pm}=n)$ as $n\rightarrow\infty.