论文标题
虚拟马尔可夫链的零法律
A Zero-One Law for Virtual Markov Chains
论文作者
论文摘要
虚拟马尔可夫链（VMC）是一个序列$ \ {x_n \} _ {n = 0}^{\ infty} $的马尔可夫链（MCS）在相同的概率空间上耦合在一起，以至于$ x_n $具有状态空间$ \ \ {0,1，\ ldots $ $ $ $ $ $ $ y y y; $ x_ {n+1}的$几乎可以肯定地在$ x_n $的样本路径中产生。在本文中，我们证明了$σ$ -Algebra $ \ bigCap_ {n = 0}^{\ infty}σ（x_n，x_n，x_ {n+1}，\ ldots）$的琐事的精确表征。这样做的主要工具是分解定理，即VMC生成的$σ$ - 代数等于某些无数无限的独立组成型MC所产生的$σ$ -Algebra。这些成分是所谓的楼梯MC（SMC），其定义为非负整数上的非苯基马尔可夫链，仅通过保持或跳到等于当前索引的值而过渡或跳过。我们还发展了SMC理论的一些一般方面，包括与凸分析的一些经典但被低估的方面的联系。
A virtual Markov chain (VMC) is a sequence $\{X_N\}_{N=0}^{\infty}$ of Markov chains (MCs) coupled together on the same probability space such that $X_N$ has state space $\{0,1,\ldots, N\}$ and such that removing all instances of $N~+~1$ from the sample path of $X_{N+1}$ results in the sample path of $X_N$ almost surely. In this paper, we prove an exact characterization of the triviality of the $σ$-algebra $\bigcap_{N=0}^{\infty}σ(X_N,X_{N+1},\ldots)$. The main tool for doing this is a decomposition theorem that the $σ$-algebra generated by a VMC is equal to the $σ$-algebra generated by a certain countably infinite collection of independent constituent MCs. These constituents are so-called staircase MCs (SMCs), which are defined to be inhomoheneous Markov chains on the non-negative integers which transition only by holding or by jumping to a value equal to the current index. We also develop some general aspects of the theory of SMCs, including a connection with some classical but very much under-appreciated aspects of convex analysis.
