论文标题
马尔可夫连锁店的零法律
A Zero-One Law for Markov Chains
论文作者
论文摘要
我们证明了对同质和非均匀马尔可夫链（MC）的经典零法则的类似物。它的几乎精确的配方很简单：鉴于任何事件$ a $从尾巴$σ$ -SALGEBRA的MC $（z_n）$的algebra（对于大$ n $）中，概率接近一个，MC的轨迹位于州$ i $中，其中$ p（a | z_n = i）$接近$ 0 $ $ 0 $或接近$ 1 $。当$ n $倾向于$ - \ infty $时，入口$σ$ -Algebra也有类似的说明。为了制定第二个结果，我们给出了有关在有限和可计数案例中以$ \ mathbb z _- $或$ \ Mathbb z $索引的非均匀马尔可夫链索引的详细结果。由于Kolmogorov，这扩展了众所周知的结果。此外，在我们的讨论中，我们注意到了MC的两个常用定义之间有趣的二分法。
We prove an analog of the classical Zero-One Law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event $A$ from the tail $σ$-algebra of MC $(Z_n)$, for large $n$, with probability near one, the trajectories of the MC are in states $i$, where $P(A|Z_n=i)$ is either near $0$ or near $1$. A similar statement holds for the entrance $σ$-algebra, when $n$ tends to $-\infty$. To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by $\mathbb Z_-$ or $\mathbb Z$ in both the finite and countable cases. This extends a well-known result due to Kolmogorov. Further, in our discussion, we note an interesting dichotomy between two commonly used definitions of MCs.
