Reliability estimation in Markov chains

In this paper, we shall study a homogeneous Markov chain with unknown transition matrix. We construct the maximum likelihood estimator of the transition probability matrix, based on the observed data and we also define estimators of availability, reliability and failure rate. Our aim is to show that these estimators are uniformly strongly consistent and that the estimators of availability and reliability converge in distribution to a normal random variable. The construction of the confidence intervals for availability and reliability are also given. Finally a numerical example for illustration is given