Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. (English) Zbl 1129.60069
The authors investigate asymptotics of moments and the weak limiting behavior of the total branch length \(L_n\) of the Bolthausen-Sznitman coalescent as the sample size \(n\) tends to \(\infty\). By using a distributional recursion that the marginal distributions of \(\{L_n: n=2,3,\ldots\}\) satisfy, a remarkable fact is proved: two term asymptotic expansions of moments \(EL_n^j\), \(j=1,2,\ldots\) coincide with those of \(EX_n^j\), \(j=1,2,\ldots\) (obtained earlier in [A. Panholzer, Mathematics and computer science III. Algorithms, trees, combinatorics and probabilities. Proceedings of the international colloquium of mathematics and computer sciences, 267–280 (2004; Zbl 1060.05022)]), where \(X_n\) is the number of collisions that take place in the restricted coalescent until there is just a single block. In a different, still unpublished paper [see also A. Iksanov and M. Möhle, Electron. Commun. Probab. 12, 28–35 (2007; Zbl 1133.60012)] the same authors prove that properly normalized and centered \(X_n\) weakly converge to a \(1\)-stable distribution. The observation that \(L_n\) and \(X_n\) are approximately the same is pushed further which allows the authors to prove that the weak asymptotic behavior of \(L_n\) coincides with that of \(X_n\).
Reviewer: Oleg K. Zakusilo (Kyïv)
MSC:
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60F05 | Central limit and other weak theorems |
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