Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent. (English) Zbl 1328.60173

Möhle, Martin

J. Appl. Probab. 51A, Spec. Vol., 87-97 (2014).

Summary: The probability \(h(n, m)\) that the block counting process of the Bolthausen-Sznitman \(n\)-coalescent ever visits the state \(m\) is analyzed. It is shown that the asymptotic hitting probabilities \(h(m) = \lim_{n\to\infty}h(n, m)\), \(m \in \mathbb N\), exist and an integral formula for \(h(m)\) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that \(h(m) \sim 1/\log m\) as \(m \to \infty\). An application to linear recursions is indicated.

MSC:

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

Keywords:

Bolthausen-Sznitman coalescent; Markov chain; asymptotic hitting probability; generating function

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