Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias

Behzad Mehrdad, Sanchayan Sen, and Lingjiong Zhu

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Abstract

Consider biased random walks on two Galton–Watson trees without leaves having progeny distributions $P_{1}$ and $P_{2}$ ($\mathrm{GW}(P_{1})$ and $\mathrm{GW}(P_{2})$) where $P_{1}$ and $P_{2}$ are supported on positive integers and $P_{1}$ dominates $P_{2}$ stochastically. We prove that the speed of the walk on $\mathrm{GW}(P_{1})$ is bigger than the same on $\mathrm{GW}(P_{2})$ when the bias is larger than a threshold depending on $P_{1}$ and $P_{2}$. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).

Résumé

Nous considérons des marches aléatoires biaisées sur deux arbres de Galton–Watson sans feuilles $\mathrm{GW}(P_{1})$ et $\mathrm{GW}(P_{2})$ ayant des lois de reproduction respectivement $P_{1}$ et $P_{2}$, deux lois supportées par les entiers positifs telles que $P_{1}$ domine stochastiquement $P_{2}$. Nous prouvons que la vitesse de la marche sur $\mathrm{GW}(P_{1})$ est supérieure ou égale á celle sur $\mathrm{GW}(P_{2})$ si le biais est plus grand qu’un seuil dépendant de $P_{1}$ et $P_{2}$. Ceci répond partiellement á une question posée par Ben Arous, Fribergh et Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 1 (2015), 304-318.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244407

Digital Object Identifier
doi:10.1214/13-AIHP573

Mathematical Reviews number (MathSciNet)
MR3300972

Zentralblatt MATH identifier
1314.60160

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

Keywords
Random walk in random environment Galton–Watson tree Speed Stochastic domination

Citation

Mehrdad, Behzad; Sen, Sanchayan; Zhu, Lingjiong. The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 304--318. doi:10.1214/13-AIHP573. https://projecteuclid.org/euclid.aihp/1421244407


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References

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