TY - JOUR

T1 - Stability of the overshoot for Lévy processes

AU - Doney, R. A.

AU - Maller, R. A.

PY - 2002

Y1 - 2002

N2 - We give equivalences for conditions like X(T(r))/r → 1 and X(T*(r))/r → 1, where the convergence is in probability or almost sure, both as r → 0 and r → ∞, where X is a Lévy process and T(r) and T*(r) are the first exit times of X out of the strip {(t, y): t > 0, |Y| ≤ r} and half-plane {(t, y): t > 0, y ≤ r}, respectively. We also show, using a result of Kesten, that X(T*(r))/r → 1 a.s. as r → 0 is equivalent to X "creeping" across a level.

AB - We give equivalences for conditions like X(T(r))/r → 1 and X(T*(r))/r → 1, where the convergence is in probability or almost sure, both as r → 0 and r → ∞, where X is a Lévy process and T(r) and T*(r) are the first exit times of X out of the strip {(t, y): t > 0, |Y| ≤ r} and half-plane {(t, y): t > 0, y ≤ r}, respectively. We also show, using a result of Kesten, that X(T*(r))/r → 1 a.s. as r → 0 is equivalent to X "creeping" across a level.

KW - Exit times

KW - First passage times

KW - Local behavior

KW - Processes with independent increments

UR - http://www.scopus.com/inward/record.url?scp=0035993023&partnerID=8YFLogxK

U2 - 10.1214/aop/1020107765

DO - 10.1214/aop/1020107765

M3 - Article

SN - 0091-1798

VL - 30

SP - 188

EP - 212

JO - Annals of Probability

JF - Annals of Probability

IS - 1

ER -