Antiduality and Möbius monotonicity: generalized coupon collector problem

Pawel Lorek

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 (FSST) 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 the limiting
distribution of the absorption time – we indicate separation cutoffs (with their window sizes) in several
chains. We also present a chain which (under some conditions) has a prescribed stationary distribution
and its FSST is distributed as a prescribed mixture of sums of geometric random variables.