Choi Hye Soo, Evans Steven N
Department of Statistics #3860, 367 Evans Hall, University of California, Berkeley, CA 94720-3860, USA.
Stoch Process Their Appl. 2017 Jul;127(7):2428-2445. doi: 10.1016/j.spa.2016.11.006. Epub 2016 Dec 5.
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 2 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 and thereby delineate all the ways in which the Markov chain can be conditioned to behave at large times. 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 bijective correspondence between the points in the boundary and ergodic random total orders on the set {, , , , …} that have distributions which are separately invariant under finite permutations of the indices of the 's and those of the 's. We establish a further bijective correspondence between the set of such random total orders and the set of pairs (, ) of diffuse probability measures on [0,1] such that ½( + ) is Lebesgue measure: the restriction of the random total order to {, ,…, } is obtained by taking ,…, (resp. ,… , ) i.i.d. with common distribution (resp. ), letting (,…, ) be {, ,…, , } in increasing order, and declaring that the smallest element in the restricted total order is (resp. ) if = (resp. = ).
我们考虑一个马尔可夫链，它以如下方式迭代生成一系列随机有限字：该字在长度为2的字的集合上均匀分布，其中在每一步， 个字母为 ， 个字母为 ，并且一个 和一个 以均匀随机的方式在当前字的字母中混洗。我们得到了这个马尔可夫链的杜布 - 马丁边界的具体特征，从而描绘出马尔可夫链在长时间条件下可能的所有行为方式。记有限字 中的字母数量（等同于 ）为 ，我们证明，如果对于任意字 ，当 趋于无穷时，从 中均匀随机选择 个字母 和 个字母 并保持它们的相对顺序得到 的概率收敛，则有限字序列( )收敛到边界中的一个点。我们展示了边界中的点与集合{, ,, ,, …}上的遍历随机全序之间的双射对应关系，这些全序的分布在 的索引和 的索引的有限置换下分别不变。我们还建立了这样的随机全序集与[0,1]上的扩散概率测度对(, )的集合之间的另一个双射对应关系，使得 ½( + )是勒贝格测度：随机全序对{, ,, …, }的限制是通过取 ,…, （分别地， ,…, ）独立同分布，共同分布为 （分别地， ），让(,…, )是{, ,, …, , }按升序排列，并声明如果 = （分别地， = ），则受限全序中第 小的元素是 （分别地， ）得到的。
