Local limit theorems for random walks conditioned to stay positive

Let $S_{0}=0,\{S_{n},\,n\geq 1\}$ be a random walk generated by a sequence of i.i.d. random variables $X_{1},X_{2},...$ and let $\tau ^{-}=\min \{n\geq 1:S_{n}\leq 0\}$ and $\tau ^{+}=\min \{n\geq 1:S_{n}>0\}$. Assuming that the distribution of $X_{1}$ belongs to the domain of attraction of an $\alpha $ -stable law we study the asymptotic behavior, as $n\rightarrow \infty $, of the local probabilities $\mathbf{P}(\tau ^{\pm }=n)$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities $ \mathbf{P}(S_{n}\in \lbrack x,x+\Delta )|\tau ^{-}>n)$ with fixed $\Delta $ and $x=x(n)\in (0,\infty )$ $.$