For hidden Markov models, one of the most popular estimates of the hidden chain is the Viterbi path—the path maximising the posterior probability. We consider a more general setting, called the pairwise Markov model (PMM), where the joint process consisting of finite-state hidden process and observation process is assumed to be a Markov chain. It has been recently proven that under some conditions the Viterbi path of the PMM can almost surely be extended to infinity, thereby defining the infinite Viterbi decoding of the observation sequence, called the Viterbi process. This was done by constructing a block of observations, called a barrier, which ensures that the Viterbi path goes through a given state whenever this block occurs in the observation sequence. In this paper, we prove that the joint process consisting of Viterbi process and PMM is regenerative. The proof involves a delicate construction of regeneration times which coincide with the occurrences of barriers. As one possible application of our theory, some results on the asymptotics of the Viterbi training algorithm are derived.

中文翻译：

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成对马尔可夫模型的维特比过程的可再生性

对于隐马尔可夫模型，最流行的隐链估计之一是维特比路径——最大化后验概率的路径。我们考虑一个更一般的设置，称为成对马尔可夫模型（PMM），其中由有限状态隐藏过程和观察过程组成的联合过程被假定为马尔可夫链。最近已经证明，在某些条件下，PMM 的维特比路径几乎可以肯定地扩展到无穷大，从而定义了观察序列的无穷维特比解码，称为维特比过程。这是通过构建一个称为屏障的观察块来完成的，它确保每当这个块出现在观察序列中时，维特比路径都会通过给定的状态。在本文中，我们证明了由维特比过程和 PMM 组成的联合过程是可再生的。证明涉及与障碍的发生一致的再生时间的精细构造。作为我们理论的一种可能应用，推导出了维特比训练算法的渐近性的一些结果。