We consider a sequence of renewal processes constructed from a sequence of random variables belonging to the domain of attraction of a stable law (1 < α < 2). We show that this sequence is not tight in the Skorokhod J₁ topology but the convergence of some functionals of it is derived. Using the structure of the sample paths of the renewal process we derive the convergence in the Skorokhod M₁ topology to an α-stable Lévy motion. This example leads to a weaker notion of weak convergence. As an application, we present limit theorems for multiple channel queues in heavy traffic. The convergence of the queue length process to a linear combination of α-stable Lévy motions is derived.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am30-1-4,
author = {Zbigniew Michna},
title = {$\alpha$-stable limits for multiple channel queues in heavy traffic},
journal = {Applicationes Mathematicae},
volume = {30},
year = {2003},
pages = {55-68},
zbl = {1026.60108},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am30-1-4}
}
Zbigniew Michna. α-stable limits for multiple channel queues in heavy traffic. Applicationes Mathematicae, Tome 30 (2003) pp. 55-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am30-1-4/