Department of Mathematics

A unified approach to heavy traffic for asymptotically stable random walks

**Vsevolod Shneer**

(Eurandom)

22 April 2009

For families of random walks {*S*_{k}^{(a)}} with **E***S*_{k}^{(a)} *= -ka < 0*
we consider their maxima M^{(a)} = sup_{k ≥ 0} *S*_{k}^{(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 the 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 will also be discussed during the talk. This is a
joint work with Vitali Wachtel (University of Munich).

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Alexandra Babenko