On Strictly Monotone Markov Chains with Constant Hitting Probabilities and Applications to a Class of Beta Coalescents

摘要

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 /3(3,6)-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,...)) is asymptotically independent as n → ∞ with X_(n,k) asymptotically Poisson distributed with parameter b/(k - 1).