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

#### M. Moehle

2018, v.24, №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