The stability of conditional Markov processes and Markov chains in random environments
The stability of conditional Markov processes and Markov chains in random environments
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随机环境中条件马尔可夫过程和马尔可夫链的稳定性
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发表时间：
2008
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通讯作者：
R. Handel
中科院分区：
文献类型：
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作者：
R. Handel
我们考虑一个离散时间隐马尔可夫模型的信号是一个平稳的马尔可夫链。当以观测值为条件时，信号是条件测度下随机环境中的马尔可夫链。证明了当信号是遍历的且观测值是非退化的时，该条件信号是弱遍历的。这允许σ-域的交集和上确界的微妙交换，这是非线性滤波器的稳定性的关键，并且部分解决了Kunita [J. Multivariate Anal. 1（1971）365 - 393]。在连续时间设置中也获得类似的结果。证明是基于一般状态空间中的随机环境中的马尔可夫链的遍历定理。
We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365―393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space.