Abstract
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.
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