Limit distributions of directionally reinforced random walks
From the introduction and the summary: The idea of reinforced random walks goes back to Coppersmith and Diaconis (1987). The recurrence properties were investigated by \textit{R. Pemantle} [Ann. Probab. 16, No. 3, 1229-1241 (1988; Zbl 0648.60077)] for the walks on an infinite tree and by \textit{B. Davis} [Probab. Theory Relat. Fields 84, No. 2, 203-229 (1990; Zbl 0665.60077)] for walks driven by simple dynamics which depend on the entire process history. As elementary mathematical models for ocean waves, \textit{R. D. Mauldin}, \textit{M. Monticino}, and \textit{H. von Weizsäcker} [Adv. Math. 117, No. 2, 239-252 (1996; Zbl 0845.60070)] recently introduced directionally reinforced random-walks in \(R^d\) which simulate some of the time and space correlations observed in ocean surface fields. The walk starts at the origin and chooses an initial direction at random. It moves in this direction with a constant speed and then changes direction at a random time. Once a new direction is chosen, the earlier directions are forgotten. Some basic properties such as recurrence properties of these walks were discussed by Mauldin et al. (loc. cit.). The main purpose of this paper is to develop the limit distributions, the strong law of large numbers, and strong approximations for the directionally reinforced random walks. 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.
scientific article; zbMATH DE number 1203781
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