Very fat geometric Galton-Watson trees
ESAIM: Probability and Statistics, Tome 24 (2020), pp. 294-314

Let τ$$ be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Z$$ = a$$} where Z$$ is the size of the nth generation and $$ is a deterministic positive sequence. We study the local limit of these trees τ$$ as n →∞ and observe three distinct regimes: if $$ grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence $$ increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.

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DOI : 10.1051/ps/2019026
Classification : 60J80, 60F15, 05C05
Keywords: Galton-Watson tree, random discrete tree, local limit, non extinction, branching process, geometric Galton-Watson process 
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     author = {Abraham, Romain and Bouaziz, Aymen and Delmas, Jean-Fran\c{c}ois},
     title = {Very fat geometric {Galton-Watson} trees},
     journal = {ESAIM: Probability and Statistics},
     pages = {294--314},
     year = {2020},
     publisher = {EDP Sciences},
     volume = {24},
     doi = {10.1051/ps/2019026},
     mrnumber = {4126978},
     zbl = {1455.60112},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2019026/}
}
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Abraham, Romain; Bouaziz, Aymen; Delmas, Jean-François. Very fat geometric Galton-Watson trees. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 294-314. doi: 10.1051/ps/2019026

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