arXiv 2202.026382022-02-05math.PR

虚马尔可夫链的零一律

A Zero-One Law for Virtual Markov Chains

Adam Quinn Jaffe

备注：: 24 pages, 3 figures

摘要：

虚拟马尔可夫链（VMC）是在相同概率空间上耦合在一起的马尔可夫链（MC）的序列$\{X_N\}_{N=0}^{\infty}$，使得$X_N$具有状态空间$\{0,1,\ldots, N\}$，并且使得从$X_{N+1}$的样本路径中移除$N~+~1$的所有实例几乎必然导致$X_N$的样本路径。本文证明了$σ$ -代数$\bigcap_{N=0}^{\infty}σ(X_N,X_{N+1},\ldots)$平凡性的一个精确刻画.这样做的主要工具是一个分解定理，即由VMC生成的$σ$ -代数等于由独立成分MC的某个可数无限集合生成的$σ$ -代数。这些成分是所谓的阶梯MC（SMC），其被定义为非负整数上的非齐次马尔可夫链，其仅通过保持或跳跃到等于当前索引的值来转移。我们还开发了SMCs理论的一些一般方面，包括与凸分析的一些经典但非常不受重视的方面的连接。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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