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

TL;DR

This paper analyzes the asymptotic behavior of the maximum of asymptotically stable random walks in heavy-traffic conditions, providing two elementary proofs and a generalized Kolmogorov inequality for infinite variance cases.

Contribution

It offers two simple proofs of the heavy-traffic limit theorem for the maximum of asymptotically stable random walks and introduces a generalized Kolmogorov inequality for infinite variance scenarios.

Findings

Two elementary proofs of the main heavy-traffic result.

A generalized Kolmogorov inequality for infinite variance.

Insights into the technical similarities of the proofs.

Abstract

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.