# Directionally Reinforced Random Walks

@article{Mauldin1996DirectionallyRR, title={Directionally Reinforced Random Walks}, author={R. Daniel Mauldin and Michael Monticino and Heinrich v. Weizs{\"a}cker}, journal={Advances in Mathematics}, year={1996}, volume={117}, pages={239-252} }

Abstract This paper introduces and analyzes a class of directionally reinforced random walks. The work is motivated by an elementary model for time and space correlations in ocean surface wave fields. We develop some basic properties of these walks. For instance, we investigate recurrence properties and give conditions under which the limiting continuous versions of the walks are Gaussian diffusion processes.

## 29 Citations

Limit Distributions of Directionally Reinforced Random Walks

- Mathematics
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Abstract As a mathematical model for time and space correlations in ocean surface wave fields, Mauldin, Monticino, and von Weizsacker (1996,Adv. in Math.117, 239–252), introduced directionally…

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This paper analyzes optimal single and multiple stopping rules for a class of correlated random walks that provides an elementary model for processes exhibiting momentum or directional reinforcement…

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- MathematicsJournal of mathematical analysis and applications
- 2021

## References

SHOWING 1-10 OF 11 REFERENCES

Principles Of Random Walk

- Mathematics
- 1964

This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of…

Phase transition in reinforced random walk and RWRE on trees

- Mathematics
- 1988

On donne a une marche aleatoire sur un arbre infini une sorte particuliere de retroaction positive de telle sorte que les aretes deja traversees aient plus de chances d'etre traversees dans le futur.…

On recurrence of a random walk in the plane

- Mathematics
- 1980

The purpose of this note is to establish a sufficient condition for recurrence of a random walk (Se) in R2. It follows from it that if SI/nn/2 is asymptotically normal then we have recurrence. Let…

A Course in Probability Theory

- Economics
- 1949

This edition of A Course in Probability Theory includes an introduction to measure theory that expands the market, as this treatment is more consistent with current courses.

Note on a class of one-dimensional reinforced r andom walks (preprint)

- Note on a class of one-dimensional reinforced r andom walks (preprint)
- 1991

On the statistical distribution of the heights of sea waves

- Journal of Marine Research
- 1952

On recurrence o f a r andom walk in the plane, P r o c . o f the

- AMS
- 1980

Reinforced r andom walk

- Probab. Th. Rel. Fields
- 1990