The natural analogue for a Lévy process of Cramér’s estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this estimate for any Lévy process with finite negative mean which satisfies Cramér’s condition, and give an explicit formula for the limiting constant. Just as in the random walk case, this leads to a Poisson limit theorem for the number of “high excursions.”
Publié le : 2005-05-14
Classification:
Maximum of reflected process,
maximal segmental score,
Poisson limit theorem,
high excursions,
60G51,
60G17
@article{1115137981,
author = {Doney, R. A. and Maller, R. A.},
title = {Cram\'er's estimate for a reflected L\'evy process},
journal = {Ann. Appl. Probab.},
volume = {15},
number = {1A},
year = {2005},
pages = { 1445-1450},
language = {en},
url = {http://dml.mathdoc.fr/item/1115137981}
}
Doney, R. A.; Maller, R. A. Cramér’s estimate for a reflected Lévy process. Ann. Appl. Probab., Tome 15 (2005) no. 1A, pp. 1445-1450. http://gdmltest.u-ga.fr/item/1115137981/