Trickle-down processes and their boundaries
Trickle-down processes and their boundaries
复制标题
滴流过程及其边界
DOI：
10.1214/ejp.v17-1698
复制
发表时间：
2010
影响因子：
1.4
通讯作者：
A. Wakolbinger
中科院分区：
文献类型：
--
作者：
S. Evans;R. Gruebel;A. Wakolbinger
可以将许多马尔可夫链中的每一个表示为一个有向无环图的连接子集的进化序列，该序列以以下方式增长：最初，图的所有顶点都是空的，粒子在一个不同的源顶点一个接一个地输入，连续的粒子根据适当的随机机制沿着有向边前进，每个粒子一旦遇到一个空顶点就会停止。例子包括二叉树和数字搜索树过程，随机递归树过程及其由Pitman的双参数中餐馆过程的嵌套实例产生的推广，与malallows的$\phi$随机排列模型和Schutzenberger的非交换$q$ -二项式定理相关的树木生长模型，以及由Luczak和Winkler提出的以马尔可夫方式生长均匀随机二叉树的构造。我们引入了一个包含这样的马尔可夫链的框架，并通过详细分析它们的Doob-Martin紧化、泊松边界和尾$\sigma$ -场来描述它们的渐近行为。
It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman's two-parameter Chinese restaurant process, tree-growth models associated with Mallows' $\phi$ model of random permutations and with Schutzenberger's non-commutative $q$-binomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail $\sigma$-fields.