The Number of Generations Entirely Visited for Recurrent Random Walks in a Random Environment

P. Andreoletti () and P. Debs
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P. Andreoletti: Université d’Orléans
P. Debs: Université d’Orléans

Journal of Theoretical Probability, 2014, vol. 27, issue 2, 518-538

Abstract: Abstract In this paper we deal with a random walk in a random environment on a super-critical Galton–Watson tree. We focus on the recurrent cases already studied by Hu and Shi (Ann. Probab. 35:1978–1997, 2007; Probab. Theory Relat. Fields 138:521–549, 2007), Faraud et al. (Probab. Theory Relat. Fields, 2011, in press), and Faraud (Electron. J. Probab. 16(6):174–215, 2011). We prove that the largest generation entirely visited by these walks behaves like logn, and that the constant of normalization, which differs from one case to another, is a function of the inverse of the constant of Biggins’ law of large numbers for branching random walks (Biggins in Adv. Appl. Probab. 8:446–459, 1976).

Keywords: Random walks; Random environment; Trees; 60J55; 60J80; 60G50; 60K37 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-012-0449-9

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