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A central limit theorem for biased random walks on Galton–Watson trees
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  • Published: 26 June 2007

A central limit theorem for biased random walks on Galton–Watson trees

  • Yuval Peres1,2 &
  • Ofer Zeitouni3,4 

Probability Theory and Related Fields volume 140, pages 595–629 (2008)Cite this article

  • 242 Accesses

  • 31 Citations

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Abstract

Let \({\mathcal{T}}\) be a rooted Galton–Watson tree with offspring distribution {p k } that has p 0 = 0, mean m = ∑ kp k > 1 and exponential tails. Consider the λ-biased random walk {X n } n ≥ 0 on \({\mathcal{T}}\) ; this is the nearest neighbor random walk which, when at a vertex v with d v offspring, moves closer to the root with probability λ/(λ + d v ), and moves to each of the offspring with probability 1/(λ + d v ). It is known that this walk has an a.s. constant speed \({\tt v} = \lim_n |X_n|/n\) (where |X n | is the distance of X n from the root), with \({\tt v} > 0\) for 0 < λ < m and \({\tt v} = 0\) for λ ≥ m. For all λ ≤ m, we prove a quenched CLT for \(|X_n| - n{\tt v}\) . (For λ >  m the walk is positive recurrent, and there is no CLT.) The most interesting case by far is λ = m, where the CLT has the following form: for almost every \({\mathcal{T}}\) , the ratio \(|X_{[nt]}|/\sqrt{n}\) converges in law as n → ∞ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for λ = 1) and the construction of appropriate harmonic coordinates.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Department of Statistics, University of California, Berkeley, USA

    Yuval Peres

  2. Microsoft Research 1 Microsoft Way, Redmond, WA, 98052-6399, USA

    Yuval Peres

  3. Department of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, 55455, USA

    Ofer Zeitouni

  4. Departments of Mathematics and of Electrical Engineering, Technion, Haifa, Israel

    Ofer Zeitouni

Authors
  1. Yuval Peres
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  2. Ofer Zeitouni
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Corresponding author

Correspondence to Ofer Zeitouni.

Additional information

Y. Peres research was partially supported by MSRI and by NSF grants #DMS-0104073 and #DMS-0244479. O. Zeitouni research was partially supported by MSRI and by NSF grants #DMS-0302230 and DMS-0503775.

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Peres, Y., Zeitouni, O. A central limit theorem for biased random walks on Galton–Watson trees. Probab. Theory Relat. Fields 140, 595–629 (2008). https://doi.org/10.1007/s00440-007-0077-y

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  • Received: 24 June 2006

  • Revised: 25 April 2007

  • Published: 26 June 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0077-y

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Mathematics Subject classification (2000)

  • Primary: 60K37
  • 60F05
  • Secondary: 60J80
  • 82C41
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