[Paper Review] Heavy-traffic analysis of the maximum of an asymptotically stable random walk

Seva Shneer, Vitali Wachtel|arXiv (Cornell University)|Feb 12, 2009

Advanced Queuing Theory Analysis12 references7 citations

TL;DR

This paper analyzes the heavy-configuration limit of the maximum of an asymptotically stable random walk as the drift parameter a approaches zero, corresponding to a queueing system under heavy traffic. Using two classical approaches—Kingman’s and Prokhorov’s—it provides elementary proofs by generalizing the Kolmogorov inequality to infinite variance settings, establishing the limiting distribution of the maximum as a stable law.

ABSTRACT

Abstract. For families of random walks {S (a) k} with ES(a) k = −ka < 0 we consider their maxima M (a) = supk≥0 S (a) k. 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. Assume that {Xi} ∞ i=1 is a sequence of i.i.d. random variables with a zero expectation: EX1 = 0. Define a random walk S0 = 0, Sk = k∑ Xi for k ≥ 1. i=1 Along with the random walk {Sk}, for each a> 0 define a random walk {S (a)} via Now we can define

Motivation & Objective

• To investigate the asymptotic behavior of the maximum of a family of asymptotically stable random walks as the drift parameter a → 0.
• To resolve the heavy-traffic approximation problem in queueing theory using elementary proofs based on Kingman’s and Prokhorov’s approaches.
• To generalize the classical Kolmogorov inequality to the case of infinite variance for i.i.d. random variables with zero mean.
• To establish the limiting distribution of the maximum of the random walk under heavy traffic, showing convergence to a stable law.

Proposed method

• Define a family of random walks {S(a)k} with negative drift −ka and zero mean increments, and consider their supremum M(a) = supk≥0 S(a)k.
• Apply Kingman’s approach by analyzing the tail behavior of M(a) through exponential martingale techniques and Laplace transform methods.
• Apply Prokhorov’s approach by using weak convergence arguments and the convergence of finite-dimensional distributions to the stable process.
• Generalize the Kolmogorov inequality to infinite variance settings by deriving moment bounds for partial sums of i.i.d. random variables with zero mean and infinite second moment.
• Use the regular variation properties of the stable distribution to characterize the limiting behavior of M(a) as a → 0.
• Establish the convergence in distribution of M(a) to a stable law with index α ∈ (0,2), under appropriate normalization.

Experimental results

Research questions

• RQ1 How does the maximum of an asymptotically stable random walk behave as the drift parameter a approaches zero?
• RQ2 Can the classical heavy-traffic approximation for queueing systems be rigorously established using elementary proofs based on Kingman’s and Prokhorov’s methods?
• RQ3 What is the appropriate generalization of the Kolmogorov inequality when the variance of the increments is infinite?
• RQ4 What is the limiting distribution of the maximum of the random walk in the heavy-traffic regime?
• RQ5 How do the technical challenges in both approaches compare, and can they be resolved via a unified inequality framework?

Key findings

• The maximum M(a) of the random walk S(a)k converges in distribution to a stable law with index α ∈ (0,2) as a → 0, under appropriate normalization.
• The generalization of the Kolmogorov inequality to infinite variance settings enables uniform moment bounds for partial sums of i.i.d. random variables with zero mean and infinite second moment.
• The technical difficulties in both Kingman’s and Prokhorov’s approaches are shown to be structurally similar and resolved via the same generalized inequality.
• The limiting distribution of M(a) is characterized by a stable law with stability index α, reflecting the long-tailed nature of the increments.
• The convergence rate and tail behavior of M(a) are consistent with the scaling properties of stable distributions in the infinite variance regime.
• The results confirm the heavy-traffic approximation for single-server queues with heavy-tailed service times, extending classical results to the stable case.