SMILE

Stochastic Models for the Inference of Life Evolution

Bibtex

@article{lambert_asymptotic_2015,
Author = {Lambert, Amaury and Simatos, Florian},
Title = {Asymptotic {Behavior} of {Local} {Times} of {Compound}
{Poisson} {Processes} with {Drift} in the {Infinite}
{Variance} {Case}},
Journal = {Journal of Theoretical Probability},
Volume = {28},
Number = {1},
Pages = {41--91},
Note = {WOS:000351316900002},
Keywords = {branching-processes, character, convergence,
Crump-Mode-Jagers branching processes, Infinite
variance, invariance-principle, Levy processes, Local
times, Processor-Sharing queue, Weak convergence},
abstract = {Consider compound Poisson processes with negative
drift and no negative jumps, which converge to some
spectrally positive L,vy process with nonzero L,vy
measure. In this paper, we study the asymptotic
behavior of the local time process, in the spatial
variable, of these processes killed at two different
random times: either at the time of the first visit of
the L,vy process to 0, in which case we prove results
at the excursion level under suitable conditionings; or
at the time when the local time at 0 exceeds some fixed
level. We prove that finite-dimensional distributions
converge under general assumptions, even if the
limiting process is not cA dlA g. Making an assumption
on the distribution of the jumps of the compound
Poisson processes, we strengthen this to get weak
convergence. Our assumption allows for the limiting
process to be a stable L,vy process with drift. These
results have implications on branching processes and in
queueing theory, namely, on the scaling limit of
binary, homogeneous Crump-Mode-Jagers processes and on
the scaling limit of the Processor-Sharing queue length
process.},
doi = {10.1007/s10959-013-0492-1},
issn = {0894-9840},
language = {English},
month = mar,
year = 2015
}