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, where the joint process consisting of finite-state hidden regime and observation process is assumed to be a Markov chain. We prove that under some conditions it is possible to extend the Viterbi path to infinity for almost every observation sequence which in turn enables to define an infinite Viterbi decoding of the observation process, called the Viterbi process. This is done by constructing a block of observations, called a barrier, which ensures that the Viterbi path goes trough a given state whenever this block occurs in the observation sequence.

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be two random sequences so that every random variable takes values in a finite set A. We consider a global similarity score Ln := L(X 1 , . . . , Xn; Y 1 , . . . , Yn) that measures the homology (relatedness) of words (X 1 , . . . , Xn) and (Y 1 , . . . , Yn). A typical example of such score is the length of the longest common subsequence. We study the order of central absolute moment E|Ln − ELn| r in the case where the two-dimensional process (X 1 , Y 1 ), (X 2 , Y 2 ), . . . is a Markov chain on A × A. This is a very general model involving independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a general condition that guarantees that E|Ln − ELn| r n r 2 . We also perform simulations indicating the validity of the condition.

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We consider a bivariate Markov chain Z = {Z k } k≥1 = {(X k , Y k )} k≥1 taking values on product space Z = X ×Y, where X is possibly uncountable space and Y = {1, . . . , |Y|} is a finite state-space. The purpose of the paper is to find sufficient conditions that guarantee the exponential convergence of smoothing, filtering and predictive probabilities:Here t ≥ s ≥ l ≥ 1, Ks is σ(Xs:∞)-measurable finite random variable and α ∈ (0, 1) is fixed. In the second part of the paper, we establish two-sided versions of the above-mentioned convergence. We show that the desired convergences hold under fairly general conditions. A special case of above-mentioned very general model is popular hidden Markov model (HMM). We prove that in HMM-case, our assumptions are more general than all similar mixing-type of conditions encountered in practice, yet relatively easy to verify.