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).
Categories: math.PR