TY - JOUR

T1 - Functionals of infinitely divisible stochastic processes with exponential tails

AU - Braverman, Michael

AU - Samorodnitsky, Gennady

N1 - Funding Information:
* Correspondinagu thor. l Braverman'rse searchw as supportedb y the U.S. Army ResearchO ffice throught he Mathematical SciencesIn stituteo f Cornell UniversityC, ontractD AAL03-91-C-0027S. amorodnitskyr'ess earchw as supportebdy the NSA Grant 92G-116a ndU nitedS tates-IsraeBli nationaSl cienceF oundation.

PY - 1995/1/1

Y1 - 1995/1/1

N2 - We investigate the tail behavior of the distributions of subadditive functionals of the sample paths of infinitely divisible stochastic processes when the Lévy measure of the process has suitably defined exponentially decreasing tails. It is shown that the probability tails of such functionals are of the same order of magnitude as the tails of the same functionals with respect to the Lévy measure, and it turns out that the results of this kind cannot, in general, be improved. In certain situations we can further obtain both lower and upper bounds on the asymptotic ratio of the two tails. In the second part of the paper we consider the particular case of Lévy processes with exponentially decaying Lévy measures. Here we show that the tail of the maximum of the process is, up to a multiplicative constant, asymptotic to the tail of the Lévy measure. Most of the previously published work in the area considered heavier than exponential probability tails.

AB - We investigate the tail behavior of the distributions of subadditive functionals of the sample paths of infinitely divisible stochastic processes when the Lévy measure of the process has suitably defined exponentially decreasing tails. It is shown that the probability tails of such functionals are of the same order of magnitude as the tails of the same functionals with respect to the Lévy measure, and it turns out that the results of this kind cannot, in general, be improved. In certain situations we can further obtain both lower and upper bounds on the asymptotic ratio of the two tails. In the second part of the paper we consider the particular case of Lévy processes with exponentially decaying Lévy measures. Here we show that the tail of the maximum of the process is, up to a multiplicative constant, asymptotic to the tail of the Lévy measure. Most of the previously published work in the area considered heavier than exponential probability tails.

KW - Exponential distributions

KW - Infinitely divisible processes

KW - Tail behavior of the distributions of functionals of sample paths

UR - http://www.scopus.com/inward/record.url?scp=0000478842&partnerID=8YFLogxK

U2 - 10.1016/0304-4149(94)00074-4

DO - 10.1016/0304-4149(94)00074-4

M3 - Article

AN - SCOPUS:0000478842

VL - 56

SP - 207

EP - 231

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 2

ER -