## Regenerative tree growth: binary self-similar continuum random trees and Poisson-Dirichlet compositions.(English)Zbl 1189.60162

Consider a stochastic binary tree growth process $$\{T_n; n\geq 1\}$$, constructed according to the following $$(\alpha,\theta)$$-selection rule: Let $$0\leq\alpha\leq 1$$ and $$\theta\geq 0$$.
(i) For $$n\geq 2$$, the tree $$T_n$$ branches at the branch point adjacent to the root into two sub-trees $$T_{n,0}$$ and $$T_{n,1}$$. Given these are of sizes $$m$$ and $$n-m$$, say, where $$T_{n,1}$$ contains the smallest label in $$T_n$$, assign the weight a to the edge connecting the root and the adjacent branch point, weights $$m-\alpha$$ and $$n-m-1+\theta$$, respectively, to the sub-trees.
(ii) Select the root edge or a sub-tree with probabilities proportional to these weights. If a sub-tree with two or more leaves was selected, recursively apply the weighting procedure (i) to the selected sub-tree, until the root edge or a sub-tree with a single leaf was selected. If a sub-tree with a single leaf was selected, select the unique edge of this sub-tree.
The limit theory of B. Haas, G. Miermont, J. Pitman and M. Winkel [Ann. Probab. 36, No. 5, 1790–1837 (2008; Zbl 1155.92033)], covers the special case $$\alpha+\theta= 1$$, but relies on sampling consistency: (Let $$T^0_n$$ be obtained from $$T_n$$ by removing the leaf labels, and $$T^0_n$$ from $$T^0_{n-1}$$ by removing a leaf chosen uniformly at random. $$\{T_n; n\geq 1\}$$ is called weakly sampling consistent if the distributions of $$T^0_n$$ and $$\widehat T^0_n$$ coincide for all $$n\geq 1$$.) In general, $$(\alpha,\theta)$$-tree growth processes are not weakly sampling consistent. So, the authors now take a new approach to the existence of compact limiting trees, which applies to the general case. It is based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Zbl 1155.92033
Full Text:

### References:

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