The Optimal Sink and the Best Source in a Markov Chain
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- Bakhtin, Y. & Bunimovich, L. J Stat Phys (2011) 143: 943. doi:10.1007/s10955-011-0223-x
It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though. Our results can be used, e.g., for a choice of the initial distribution in numerical experiments with the fastest convergence to equilibrium/stationary distribution, for characterization of the elements of a dynamical network according to their ability to absorb and transmit the substance (“information”) that is circulated over the network, for determining optimal stopping moments (stopping signals/words) when dealing with sequences of symbols, etc.