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Asymptotic properties of expansive Galton-Watson trees. (English) Zbl 1466.60171

Summary: We consider a super-critical Galton-Watson tree \(\tau \) whose non-degenerate offspring distribution has finite mean. We consider the random trees \(\tau _n\) distributed as \(\tau \) conditioned on the \(n\)-th generation, \(Z_n\), to be of size \(a_n\in{\mathbb N} \). We identify the possible local limits of \(\tau _n\) as \(n\) goes to infinity according to the growth rate of \(a_n\). 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 \(Z_n\). 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 \((\tau ^\theta , \theta \in [0, \infty ])\).

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F15 Strong limit theorems
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