The Annals of Probability

The slow regime of randomly biased walks on trees

Yueyun Hu and Zhan Shi

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Abstract

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.

Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3893-3933.

Dates
Received: October 2014
Revised: September 2015
First available in Project Euclid: 14 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1479114266

Digital Object Identifier
doi:10.1214/15-AOP1064

Mathematical Reviews number (MathSciNet)
MR3572327

Zentralblatt MATH identifier
1360.60156

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

Keywords
Biased random walk on the Galton–Watson tree branching random walk slow movement local time convergence in law

Citation

Hu, Yueyun; Shi, Zhan. The slow regime of randomly biased walks on trees. Ann. Probab. 44 (2016), no. 6, 3893--3933. doi:10.1214/15-AOP1064. https://projecteuclid.org/euclid.aop/1479114266


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