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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 $${x=x(n)\in (0,\infty )}$$.

Local probabilities for random walks conditioned to stay positive