Abstract
We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space. The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called random-free or product measurable) is simpler than the general case.
Highlights
Suppose we have a continuous time Markov chain with a very large, but finite, number of states. (We are interested in the case where the chain is reversible and time-homogeneous.) We would expect that the chain resembles a chain with an infinite measurable state space
Following a suggestion by Aldous [1], we assume the mixing of the sequence is uniformly bounded
We identify the sequence with an infinite Markov chain and show that the statistical behavior of the finite chains converges to the statistical behavior of this infinite chain
Summary
Suppose we have a continuous time Markov chain with a very large, but finite, number of states. (We are interested in the case where the chain is reversible and time-homogeneous.) We would expect that the chain resembles a chain with an infinite measurable state space. Following a suggestion by Aldous [1], we assume the mixing of the sequence is uniformly bounded (that is, for each time t there is a bound Bt such that the mixing of each chain at time t is bounded by Bt) Given such a sequence, we identify the sequence with an infinite Markov chain and show that the statistical behavior of the finite chains converges to the statistical behavior of this infinite chain. Gromov [12] identified convergent sequences of metric measure spaces with certain partially exchangeable arrays of random variables. Elek identifies each such array, essentially uniquely, with an infinitary object (a “quantum metric measure space”). In our case it allows us to avoid certain complications compared to the graph case
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