Antiduality and Möbius 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 Möbius 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.