  Let S_0=0,{S_n, n>0} be a random walk generated by a sequence of i.i.d.
random variables X_1,X_2,... and let \tau^{-} be the first descending ladder
epoch. Assuming that the distribution of X_1 belongs to the domain of
attraction of an \alpha-stable law we study the asymptotic behavior of the
local probabilities P(\tau ^{-}=n) and the conditional local probabilities
P(S_n\in [x,x+y)|\tau^{-}>n) for fixed y and x=x(n)\in (0,\infty).
