On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity. (English) Zbl 1341.60088

Möhle, Martin

ALEA, Lat. Am. J. Probab. Math. Stat. 11, No. 1, 141-159 (2014).

Summary: Let \(X = (X_k)_{k=0,1,\ldots}\) denote the jump chain of the block counting process of the \(\Lambda \)-coalescent with \(\Lambda = \beta (2 - \alpha , \alpha )\) being the beta distribution with parameter \(\alpha \in (0, 2)\). A solution for the hitting probability \(h(n, m)\) that the chain \(X\) ever visits the state \(m\), conditional that it starts in the state \(X_0 = n\), is obtained via an analytic method based on generating functions. For \(\alpha \in (1, 2)\) the results are applied to characterize the distribution of the almost sure limit \(\tau \) of the absorption times \(\tau_n\) of the coalescent restricted to a sample of size \(n\). The latter result is generalized to arbitrary exchangeable coalescents (\(\Xi\)-coalescents) that come down from infinity. The results generalize those obtained for the particular case \(\alpha = 1\) by the author [J. Appl. Probab. 51A, 87–97 (2014; Zbl 1328.60173)]. This article furthermore supplements the work of O. Hénard [Ann. Appl. Probab. 25, No. 5, 3007–3032 (2015; Zbl 1325.60124)].

MSC:

60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60C05	Combinatorial probability
05C05	Trees
92D15	Problems related to evolution

Keywords:

beta coalescents; Bolthausen-Sznitman coalescent; absorption time; generating functions; Green matrix; hitting probability

Citations:

Zbl 1328.60173; Zbl 1325.60124