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Limit Distributions of Directionally Reinforced Random Walks

✍ Scribed by Lajos Horváth; Qi-Man Shao

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 263 KB
Volume: 134
Category: Article
ISSN: 0001-8708
DOI: 10.1006/aima.1997.1707

No coin nor oath required. For personal study only.

✦ Synopsis

As a mathematical model for time and space correlations in ocean surface wave fields, Mauldin, Monticino, and von Weizsa cker (1996, Adv. in Math. 117, 239 252), introduced directionally reinforced random walks and analyzed their recurrence properties. This paper develops limiting properties for these random walks in R d . In particular, it is shown that the limit distribution is a Brownian motion if the time between changes of directions has a finite second moment, and is a stable process if the time between changes of directions is in a domain of attraction of a stable law. The latter gives an affirmative answer to an open question posed by Mauldin et al.. The strong law of large numbers and strong approximations are also discussed.

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