Antiduality and M\"obius monotonicity: Generalized Coupon Collector Problem

Abstract

For a given absorbing Markov chain $X^*$ on a finite state space, a chain $X$ is a sharp antidual of $X^*$ if the fastest strong stationary time of $X$ is equal, in distribution, to the absorption time of $X^*$. In this paper we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for M\"obius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence - utilizing known results on a limiting distribution of the absorption time - we indicate a separation cutoff (with its window size) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its fastest strong stationary time is distributed as a prescribed mixture of sums of geometric random variables.