2009/02/12 by Seva Shneer, Shneer, Seva, Vitali Wachtel +1
Business, Management and Accounting · Decision Sciences · Mathematics · #60F05 #60G50 #60K25 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60G50 #msc:60K25
paper · pdf · doi:10.48550/arxiv.0902.2185
9 pages
arxiv created 2009/02/12 · openalex publication_date 2009/02/12 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
For families of random walks \Sk(a)\ with \mathbf E Sk(a) = -ka < 0 we consider their maxima M(a) = supk ≥ 0 Sk(a). We investigate the asymptotic behaviour of M(a) as a → 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.