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(-) = min{n >= 1 : S-n <= 0} and tau(+) = min{n >= 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 -> infinity, of the local probabilities P(tau(+/-) = n) and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities P(S-n is an element of [x, x + Delta)vertical bar tau(-) > n) with fixed Delta and x = x(n) is an element of (0, infinity).