Limit theorems for the inductive mean on metric trees. (English) Zbl 1219.60019
Consider a sequence of i.i.d. random variables \((X_i)\) taking their values in a metric tree. The notion of inductive mean \((\vec X_n)\) constructed from the first \(n\) values of the sequence is considered. Under a convenient framework, \(\vec X_n\) converges to the barycenter of the law of \(X_1\). The goal of this article is to study the small deviations from this limit. More precisely, a central limit theorem is proved when the barycenter is not a vertex of the tree, but if it is a vertex, then the behaviour is different and in particular the rate of convergence is different.
Reviewer: Jean Picard (Aubière)
MSC:
| 60F05 | Central limit and other weak theorems |
| 92B10 | Taxonomy, cladistics, statistics in mathematical biology |
| 60K10 | Applications of renewal theory (reliability, demand theory, etc.) |
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