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Journal of the London Mathematical Society 1999 60(2):630-640; doi:10.1112/S0024610799008017
© 1999 by London Mathematical Society
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© The London Mathematical Society

Fastest Coupling of Random Walks

L. C. G. Rogers

School of Mathematical Sciences, University of Bath Bath BA2 7AY lcgr{at}maths.bath.ac.uk

Received 23 April 1997. Revision received 4 May 1998.

A new coupling of one-dimensional random walks is described which tries to control the coupling by keeping the separation of the two random walks of constant sign. It turns out that among such monotone couplings there is an optimal one-step coupling which maximises the second moment of the difference (assuming this is finite), and this coupling is ‘fast’ in the sense that for a random walk with a unimodal step distribution the coupling time achieved by using the new coupling at each step is stochastically no larger than any other coupling. This is applied to the case of symmetric unimodal distributions.


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