Power moments of first passage times for some oscillating perturbed random walks

B. Rashytov
Theory of Stochastic Processes
Vol.23 (39), no.1, 2018, pp.93-97
Let (ξ₁, η₁), (ξ₂, η₂) … be a sequence of i.i.d. random vectors taking values in R², and let S₀:=0 and S_n:=ξ₁+…+…ξ_n for n ∈ N. The sequence (S_n-1+η_n)_n ∈ N is then called perturbed random walk. For real x, denote by τ(x) the first time the perturbed random walk exits the interval (-∞, x]. We consider a rather intricate case in which S_n drifts to the left, yet the perturbed random walk oscillates because of occasional big jumps to the right of the perturbating sequence (η_n)_n ∈ N. Under these assumptions we provide necessary and sufficient conditions for the finiteness of power moments of τ(x), thereby solving an open problem posed by Alsmeyer, Iksanov and Meiners in [2].