Local probabilities for random walks conditioned to stay positive

TL;DR

This paper investigates the asymptotic behavior of local probabilities for a random walk conditioned to stay positive, focusing on the first descending ladder epoch and stable law domains.

Contribution

It provides new asymptotic results for local probabilities of random walks conditioned to remain positive, especially within the domain of attraction of stable laws.

Findings

Asymptotic behavior of P(τ⁻=n) characterized

Conditional local probabilities P(S_n ∈ [x, x+y]) analyzed

Results applicable to stable law domains

Abstract

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).