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Scaling limit of the random walk on a Galton-Watson tree with regular varying offspring distribution

Dongjian Qian, Yang Xiao

Abstract

We consider a random walk on a Galton-Watson tree whose offspring distribution has a regular varying tail of order $κ\in (1,2)$. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the continuum tree coded by the latter continuous-time height process.

Scaling limit of the random walk on a Galton-Watson tree with regular varying offspring distribution

Abstract

We consider a random walk on a Galton-Watson tree whose offspring distribution has a regular varying tail of order . We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the continuum tree coded by the latter continuous-time height process.
Paper Structure (8 sections, 106 equations)

This paper contains 8 sections, 106 equations.

Table of Contents

  1. Introduction
  2. Overview of the proof
  3. Reduction of trees
  4. Convergence of the height process associated with $\mathbf{F}^R$ and $\mathbf{F}^X$
  5. Change of measure
  6. Proof of Proposition 2
  7. Hypotheses ($\mathbf H_l$)(i),(ii) and (iv)
  8. Regular varying tail of $L^1$

Theorems & Definitions (8)

  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Proposition \ref{['Prop_main']}
  • proof
  • proof
  • proof : proof of Proposition \ref{['Prop_rvofL1']}