THE UNIFORM LOCAL ASYMPTOTICS OF THE OVERSHOOT OF A RANDOM WALK WITH HEAVY-TAILED INCREMENTS

摘要

The article obtains the uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. When the overshoot is in (x + y, x + y + T] for some 0 < T < ∞, we discuss the asymptotics for the cases of y ≥f(x) for any positive function f(x) → ∞ as x → ∞ and y ∈ [0, N] for any 0 < N < ∞, respectively. Based on the stated results, we get a unified form of the local asymptotics of the overshoot of a random walk for the case y ≥ 0. The obtained results require certain uniform requirements for y.