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

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 < infinity, we discuss the asymptotics for the cases of y >= f (x) for any positive function f (x) -> infinity as x -> infinity and y is an element of [0, N] for any 0 < N < infinity, 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.