Volume 23, 2019
|Page(s)||739 - 769|
|Published online||20 December 2019|
Antiduality and Möbius monotonicity: generalized coupon collector problem☆
Mathematical Institute, University of Wrocław,
pl. Grunwaldzki 2/4,
* Corresponding author: firstname.lastname@example.org
Accepted: 5 March 2019
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.
Mathematics Subject Classification: 60J10 / 60G40 / 06A06
Key words: Markov chains / strong stationary duality / antiduality / absorption times / fastest strong stationary times / Möbius monotonicity / generalized coupon collector problem / Double Dixie cup problem / separation cutoff / partial ordering / perfect simulation
© EDP Sciences, SMAI 2019
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