Local probabilities for random walks conditioned to stay positive

Let S ₀ = 0, {S _n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X ₁, X ₂, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X ₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ .