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

#### M. Moehle

2018, v.24, Issue 1, 107-130

ABSTRACT

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 $\beta(3,b)$-coalescen

t 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

$b/(k-1)$.

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

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