### Abstract

We consider a biased random walk X _{n} on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |X _{n}| is of order n ^{γ}. Denoting {increment} _{n} the hitting time of level n, we prove that {increment} _{n}/n ^{1/γ} is tight. Moreover, we show that {increment} _{n}/n ^{1/γ} does not converge in law (at least for large values of β). We prove that along the sequences n _{λ}(k) =⌊λβ ^{γk}⌋, {increment} _{n}/n ^{1/γ} converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

Original language | English (US) |
---|---|

Pages (from-to) | 280-338 |

Number of pages | 59 |

Journal | Annals of Probability |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2012 |

### Fingerprint

### Keywords

- Electrical networks
- Galton-Watson tree
- Infinitely divisible distributions
- Random walk in random environment

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*40*(1), 280-338. https://doi.org/10.1214/10-AOP620

**Biased random walks on Galton-Watson trees with leaves.** / Ben Arous, Gérard; Fribergh, Alexander; Gantert, Nina; Hammond, Alan.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 40, no. 1, pp. 280-338. https://doi.org/10.1214/10-AOP620

}

TY - JOUR

T1 - Biased random walks on Galton-Watson trees with leaves

AU - Ben Arous, Gérard

AU - Fribergh, Alexander

AU - Gantert, Nina

AU - Hammond, Alan

PY - 2012/1

Y1 - 2012/1

N2 - We consider a biased random walk X n on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |X n| is of order n γ. Denoting {increment} n the hitting time of level n, we prove that {increment} n/n 1/γ is tight. Moreover, we show that {increment} n/n 1/γ does not converge in law (at least for large values of β). We prove that along the sequences n λ(k) =⌊λβ γk⌋, {increment} n/n 1/γ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

AB - We consider a biased random walk X n on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |X n| is of order n γ. Denoting {increment} n the hitting time of level n, we prove that {increment} n/n 1/γ is tight. Moreover, we show that {increment} n/n 1/γ does not converge in law (at least for large values of β). We prove that along the sequences n λ(k) =⌊λβ γk⌋, {increment} n/n 1/γ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

KW - Electrical networks

KW - Galton-Watson tree

KW - Infinitely divisible distributions

KW - Random walk in random environment

UR - http://www.scopus.com/inward/record.url?scp=84865081425&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865081425&partnerID=8YFLogxK

U2 - 10.1214/10-AOP620

DO - 10.1214/10-AOP620

M3 - Article

AN - SCOPUS:84865081425

VL - 40

SP - 280

EP - 338

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -