Abstract: Let S(t) be a random process defined for all t ≥ 0 with S(0) = 0 and let g(t) be a non-random function on [0, ∞). Introduce the random variable τ := inf {t > 0 : S(t) ≤ g(t)}= inf {t > 0 : S(t) − g(t) ≤ 0}, equal to the first moment of the top-down crossing of the level g(t) by our process S(t). We consider in the talk the asymptotic behavior of the upper tail P(τ > T) as T → ∞. For several classes of processes we are going to obtain conditions under which we have asymptotical formulas of the following type: P(τ > T) ∼U(T)/√T as T → ∞, (1) for some slowly varying functions U(T). Two such classes of processes may be found in papers [1] and [2]. In these cases U(T) := \sqrt{2/π}E[S(α_T ) − g(α_T ) ; τ > α_T } (2) for specially chosen stopping times α_T such that α_T /T → 0 in probability as T → ∞.

Cite: Sakhanenko A.
On rst-passage times for generalized random walks
International Conference dedicated to the 90th birthday of Ildar Ibragimov 30 сент. - 2 окт. 2022