Second-Order Moments of a Stationary Markov Chain and Some Applications

摘要

The i-th state of a finite-state Markov chain can be indicated by a vector with 1 in the i-th position and O's in the other positions. The vector-valued process defined in this way is autoregressive in the wide sense. The second-order moments, moving average representation, and spectral density of this process are obtained. The numerical-valued chain is a linear function of the vector; a simple condition is derived for it to be first-order autoregressive in the wide sense. The stationary probabilities of the chain can be estimated by the means of observations on the vector-valued process; this estimator is asymptotically equivalent to the maximum likelihood estimator. (JHD)