Regenerative tree growth: structural results and convergence. (English) Zbl 1304.60096

Summary: We introduce regenerative tree growth processes as consistent families of random trees with \(n\) labelled leaves, \(n\geq 1\), with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin’s notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by B. Haas and G. Miermont [Bernoulli 17, No. 4, 1217–1247 (2011; Zbl 1263.92034); Ann. Probab. 40, No. 6, 2589–2666 (2012; Zbl 1259.60033)] for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F17 Functional limit theorems; invariance principles
60G18 Self-similar stochastic processes
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