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- W4298286506 abstract "We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $tau$n distributed as $tau$ conditioned on the n-th generation, Zn, to be of size an $in$ N. We identify the possible local limits of $tau$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $tau$ 0 is the Kesten tree, in the moderate regime the family of local limits, $tau$ $theta$ for $theta$ $in$ (0, +$infty$), is distributed as $tau$ conditionally on {W = $theta$}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $tau$ $infty$ 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 ($tau$ $theta$ , $theta$ $in$ [0, $infty$])." @default.
- W4298286506 created "2022-10-01" @default.
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- W4298286506 date "2017-12-13" @default.
- W4298286506 modified "2023-09-24" @default.
- W4298286506 title "Asymptotic properties of expansive Galton-Watson trees" @default.
- W4298286506 doi "https://doi.org/10.48550/arxiv.1712.04650" @default.
- W4298286506 hasPublicationYear "2017" @default.
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