It is shown that the large-deviation statistical quantities of the discrete-time, finite-state Markov process P-n+1((j))=Sigma(k=1HjkPn(k))-H-N, where P-n((j)) is the probability for the j state at the time step n and H-jk is the transition probability, completely coincide with those from the Kalman map corresponding to the above Markov process. Furthermore, it is demonstrated that, by using simple examples, time correlation functions in finite-state Markov processes can be well described in terms of unstable periodic orbits embedded in the equivalent Kalman maps.