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Stochastic Processes and their Applications 38 (1991) 185-193 North-Holland
 

Summary: Stochastic Processes and their Applications 38 (1991) 185-193
North-Holland
185
Meeting times for independent Markov chains
David J. Aldous
Department of Statistics, Uniuersity of California, Berkeley, CA 94720, USA
Received 1 June 1988
Revised 3 September 1990
Start two independent copies of a reversible Markov chain from arbitrary initial states. Then the expected
time until they meet is bounded by a constant times the maximum first hitting time for the single chain.
This and a sharper result are proved, and several related conjectures are discussed.
1. Introduction
Let (X,) be an irreducible continuous-time pure jump Markov chain on finite state
space I = {i, j, k, . . .} with stationary distribution n: Classical theory says P(X, = j) +
3 as t -+ COfor all j, regardless of the initial distribution. The modern `coupling'
proof goes as follows. Let (Y,) be an independent copy of the chain. Then (X,, Y,),
considered as a chain on I x I, is irreducible and hence the meeting time
T,=min{t: X,= Y,}
is a.s. finite, regardless of the initial distributions. Now give Y,, the stationary
distribution and define

  

Source: Aldous, David J. - Department of Statistics, University of California at Berkeley

 

Collections: Mathematics