Picture of automobile manufacturing plant

Driving innovations in manufacturing: Open Access research from DMEM

Strathprints makes available Open Access scholarly outputs by Strathclyde's Department of Design, Manufacture & Engineering Management (DMEM).

Centred on the vision of 'Delivering Total Engineering', DMEM is a centre for excellence in the processes, systems and technologies needed to support and enable engineering from concept to remanufacture. From user-centred design to sustainable design, from manufacturing operations to remanufacturing, from advanced materials research to systems engineering.

Explore Open Access research by DMEM...

Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

MacPhee, I.M. and Menshikov, Mikhail V. and Wade, A.R. (2010) Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift. Markov Processes and Related Fields, 16 (2). pp. 351-388. ISSN 1024-2953

[img]
Preview
PDF
0910.1772v1.pdf - Accepted Author Manuscript

Download (515kB) | Preview

Abstract

We study the rst exit time from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on Zd (d 2) with mean drift that is asymptotically zero. Specically, if the mean drift at x 2 Zd is of magnitude O(kxk&#x100000;1), we show that < 1 a.s. for any cone. On the other hand, for an appropriate drift eld with mean drifts of magnitude kxk&#x100000;, 2 (0; 1), we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.