Markovljevi lanci unatrag

In this thesis, we introduce Markov chains backwards in discrete time. In the first chapter, we defined Markov chains in discrete time. Also, we have defined important terms such as transition probability matrix, time of the first visit, time of the first return, invariant measure, state graph etc. We classified Markov chain states and presented results that are important for understanding Markov chains. In the second chapter, which is the main part of this thesis, we have defined Markov chains backwards and we have proved their properties. We introduced balance equation which holds for matrices P, Pˆ and for measure λ. Later we introduced more complex balance equation so as balance equation that holds for n state transition probability matrices. Further, we set an example of Markov chain backwards. Markov chains backwards whose transition probability matrices are identical to original transition probability matrices were of special interest to us. They are known as reversible Markov chains. We also proved some important properties of reversible Markov chains. Furthermore, we discussed reversibility of Ehrenfest chain, reversibility of Bernoulli-Laplace chain and reversibility of birth and death chains in separate subsections. Also, we discussed random walks on graphs in terms of reversible Markov chains and birth- death chains in terms of Markov chains backwards