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Chapter 8: Markov Chains - Auckland

149 Chapter 8: Markov far, we have examined several stochastic processes usingtransition diagrams and First-Step processes can be written as{X0, X1, X2, ..},whereXtis thestate at the transition diagram,Xtcorresponds towhich box we are in at the Gambler s Ruin (Section ),Xtis the amount of money the gamblerpossesses after tosst. In the model for gene spread (Section ),Xtis thenumber of animals possessing the harmful allele A in processes that we have looked at via the transition diagram have a crucialproperty in common:Xt+1depends only does notdepend uponX0, X1.

151 8.2 Definitions The Markov chain is the process X 0,X 1,X 2,.... Definition: The state of a Markov chain at time t is the value ofX t. For example, if X t = 6, we say the process is in state6 at timet. Definition: The state space of a Markov chain, S, is the set of values that each

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