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We begin with the intuitive background on Markov Chains, and then talk about the necessity of positive recurrence assumption of a Markov Chain for the existence of Central Limit Theorem. Then we introduce additive functionals of countable Markov Chains and talk about two methods, namely, Dˆblin method and Martingale ap- proach, to prove the Central Limit Theorem for the ergodic Loop Markov Chains. Lastly, we introduce three Loop Markov Chain models, and construct the Central Limit Theorem for them.