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The primary goal of this book is to present a comprehensive and theoretically accessible work to graduate students and researchers to develop their skills to use the theory of statistical inference problems on Markov processes. Therefore, the main objectives of this book are to produce an expository survey of the basic results on the estimation theory and testing hypotheses in discrete Markov chains. The survey is intended to be a self- contained in the sense that the complete proofs of some important results such as the asymptotic properties of the maximum likelihood estimators, are given without resorting to sequential or cross referencing. Some tests of hypotheses for absorbing chains are also reviewed. Also, to carryout some minor contributions through the generalization and extension of certain results.