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 -