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Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

Part of: Stochastic systems and control Markov processes

Published online by Cambridge University Press:  30 January 2018

Stephen Connor
Stephen Connor*
Affiliation:
University of York
*
∗ Postal address: Department of Mathematics, University of York, York, YO10 5DD, UK. Email address: [email protected]
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Abstract

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The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

Keywords

Optimal couplingcoadaptedstochastic controlrandom walk on a groupcutoff phenomenon

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1117 - 1130
Copyright
© Applied Probability Trust 

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