Persistent Random Walks. I. Recurrence Versus Transience
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We consider a walker on the line that at each step keeps the same direction with a probability which depends on the time already spent in the direction the walker is currently moving. These walks with memories of variable length can be seen as generalizations of directionally reinforced random walks introduced in Mauldin et al. (Adv Math 117(2):239–252, 1996). We give a complete and usable characterization of the recurrence or transience in terms of the probabilities to switch the direction and we formulate some laws of large numbers. The most fruitful situation emerges when the running times both have an infinite mean. In that case, these properties are related to the behaviour of some embedded random walk with an undefined drift so that these features depend on the asymptotics of the distribution tails related to the persistence times. In the other case, the criterion reduces to a null-drift condition. Finally, we deduce some criteria for a wider class of persistent random walks whose increments are encoded by a variable length Markov chain having—in full generality—no renewal pattern in such a way that their study does not reduce to a skeleton RW as for the original model.
KeywordsPersistent and directionally reinforced random walks Variable length memory Recurrence and transience Random walk with undefined mean
Mathematics Subject Classification (2010)60G50 60J15 60G17 60J05 37B20 60K35
The authors wish to thank the referee for his or her valuable advice and suggestions – especially Remark 3.1 – improving the readability of the exposition and enriching the content of the paper.
- 2.Cénac, P., Chauvin, B., Paccaut, F., Pouyanne, N.: Context trees, variable length Markov chains and dynamical sources. In: Séminaire de Probabilités XLIV, Lecture Notes in Math., vol. 2046, pp. 1–39. Springer, Heidelberg (2012)Google Scholar
- 4.Eckstein, E.C., Goldstein, J.A., Leggas, M.: The mathematics of suspensions: Kac walks and asymptotic analyticity. In: Proceedings of the Fourth Mississippi State Conference on Difference Equations and Computational Simulations (1999), Electron. J. Differ. Equ. Conf., vol. 3, pp. 39–50. Southwest Texas State Univ., San Marcos, TX (2000)Google Scholar
- 5.Erickson, K.B.: The strong law of large numbers when the mean is undefined. Trans. Am. Math. Soc. 185, 371–381 (1974/1973)Google Scholar
- 9.Kac, M.: A stochastic model related to the telegrapher’s equation. Rocky Mountain J. Math. 4, 497–509 (1974). Reprinting of an article published in 1956, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972)Google Scholar