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

Abstract

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 $p_k={\mathbb P}\{\xi=k\}$, $k\in{\mathbb N}$. Depending on the tail behaviour of the distribution of $\xi$, 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.