On strictly monotone Markov chains with constant hitting probabilities and applications to a class of beta coalescents. (English) Zbl 1417.60063

Möhle, M.

Markov Process. Relat. Fields 24, No. 1, 107-130 (2018).

Summary: Strictly monotone Markov chains with constant hitting probabilities are characterized. The results are applied to the block counting process and the fixation line of the \(\beta(3,b)\)-coalescent with parameter \(b>0\) leading to exact convolution representations for the number of collisions, the absorption time and the total tree length of the coalescent restricted to a sample of size \(n\). The number of collisions \(X_{n,k}\) involving exactly \(k\) blocks is analyzed. The collision spectrum \((X_{n,2}, X_{n,3},\ldots)\) is asymptotically independent as \( n \to \infty\) with \(X_{n,k}\) asymptotically Poisson distributed with parameter \(b/(k-1)\).

MSC:

60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
62E15	Exact distribution theory in statistics
60F05	Central limit and other weak theorems
92D15	Problems related to evolution

Keywords:

absorption time; beta coalescent; block counting process; collision spectrum; fixation line; hitting probability; jump spectrum; monotone Markov chain; number of collisions; tree length