Local probabilities for random walks conditioned to stay positive

Details Local probabilities for random walks conditioned to stay positive Local probabilities for random walks conditioned to stay positive

2007-11-08

arxiv.org/abs/0711.1302v1

Mathematics

Probability

Scientific paper

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).

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