Heavy-traffic analysis of the maximum of an asymptotically stable random walk

Details Heavy-traffic analysis of the maximum of an asymptotically stable random walk Heavy-traffic analysis of the maximum of an asymptotically stable random walk

2009-02-12

arxiv.org/abs/0902.2185v1

Mathematics

Probability

9 pages

Scientific paper

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to 1 and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance. Such a generalisation is also obtained in this note.

Affiliated with

Also associated with

No associations

Rating

Heavy-traffic analysis of the maximum of an asymptotically stable random walk does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Heavy-traffic analysis of the maximum of an asymptotically stable random walk, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Heavy-traffic analysis of the maximum of an asymptotically stable random walk will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-556952