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Asymptotic properties of expansive Galton-Watson trees

Abstract : We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees τn distributed as τ conditioned on the n-th generation, Zn, to be of size an ∈ N. We identify the possible local limits of τn as n goes to infinity according to the growth rate of an. In the low regime, the local limit τ 0 is the Kesten tree, in the moderate regime the family of local limits, τ θ for θ ∈ (0, +∞), is distributed as τ conditionally on {W = θ}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards τ ∞ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits (τ θ , θ ∈ [0, ∞]).
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Submitted on : Tuesday, December 12, 2017 - 11:09:42 AM
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Romain Abraham, Jean-François Delmas. Asymptotic properties of expansive Galton-Watson trees. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2019, 24, Paper 15, 1-51. ⟨10.1214/19-EJP272⟩. ⟨hal-01661755⟩

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