Martin Möhle — Page 2

arXiv2008

On asymptotics of exchangeable coalescents with multiple collisions

We study the number of collisions $X_n$ of an exchangeable coalescent with multiple collisions ($Λ$-coalescent) which starts with $n$ particles and is driven by rates determined by a finite characteristic measure $ν({\rm d}x)=x^{-2}Λ({\rm d}x)$. Via a coupling technique we derive limiting laws of $X_n$, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of $X_n$ include normal, stable with index $1\leα<2$ and Mittag-Leffler distributions. The results apply, in particular, to the case when $ν$ is a beta$(a-2,b)$ distribution with parameters $a>2$ and $b>0$. The approach taken allows to derive asymptotics of three other functionals of the coalescent, the absorption time, the length of an external branch chosen at random from the $n$ external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.

math.PR↗

arXiv2007

On a random recursion related to absorption times of death Markov chains

Let $X_1,X_2,...$ be a sequence of random variables satisfying the distributional recursion $X_1=0$ and $X_n= X_{n-I_n}+1$ for $n=2,3,...$, where $I_n$ is a random variable with values in $\{1,...,n-1\}$ which is independent of $X_2,...,X_{n-1}$. The random variable $X_n$ can be interpreted as the absorption time of a suitable death Markov chain with state space ${\mathbb N}:=\{1,2,...\}$ and absorbing state 1, conditioned that the chain starts in the initial state $n$. This paper focuses on the asymptotics of $X_n$ as $n$ tends to infinity under the particular but important assumption that the distribution of $I_n$ satisfies ${\mathbb P}\{I_n=k\}=p_k/(p_1+...+p_{n-1})$ for some given probability distribution $p_k={\mathbb P}\{ξ=k\}$, $k\in{\mathbb N}$. Depending on the tail behaviour of the distribution of $ξ$, several scalings for $X_n$ and corresponding limiting distributions come into play, among them stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is a coupling technique which relates the distribution of $X_n$ to a random walk, which explains, for example, the appearance of the Mittag-Leffler distribution in this context. The results are applied to describe the asymptotics of the number of collisions for certain beta-coalescent processes.

math.PR↗