Antiduality and Möbius monotonicity: generalized coupon collector problem^☆

Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.

^* Corresponding author: pawel.lorek@math.uni.wroc.pl

Received: 28 November 2016
Accepted: 5 March 2019

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.