Doob-Martin compactification of a Markov chain for growing random words sequentially
Hye Soo Choi, Steven N. Evans

TL;DR
This paper characterizes the Doob-Martin boundary of a Markov chain that generates random words with equal numbers of 'a' and 'b', linking boundary points to invariant random total orders and pairs of probability measures.
Contribution
It provides a concrete description of the Doob-Martin boundary for a specific Markov chain and establishes bijections with invariant random orders and measure pairs.
Findings
Boundary points correspond to limits of certain probability convergence.
A bijection exists between boundary points and ergodic invariant total orders.
These orders relate to pairs of diffuse probability measures with a specific average.
Abstract
We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the word is uniformly distributed over the set of words of length in which letters are and letters are : at each step an and a are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain. Writing for the number of letters (equivalently, ) in the finite word , we show that a sequence of finite words converges to a point in the boundary if, for an arbitrary word , there is convergence as tends to infinity of the probability that the selection of letters and letters uniformly at random from and maintaining their relative order results in . We exhibit a…
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