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A series of independent events (for example, a series of coin flips) satisfies the formal definition of a Markov chain. 5 The probability of achieving [52], Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through Harris chains. Markov chains have many applications as statistical models of real-world processes,[1][4][5][6] such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. A Markov chain is a mathematical process that transitions from one state to another within a finite number of possible states. X Sign … It is not aware of its past (that is, it is not aware of what is already bonded to it). {\displaystyle X_{n}} k G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, This page was last edited on 29 November 2020, at 07:37. {\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,} use probabilistic reasoning to obtain an integral equation that the semigroup must satisfy. A discrete-time Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states: The possible values of Xi form a countable set S called the state space of the chain. such that, with in the stationary distribution on the following Markov chain on all (known) webpages. k If it ate lettuce today, tomorrow it will eat grapes with probability 4/10 or cheese with probability 6/10. [59], The paths, in the path integral formulation of quantum mechanics, are Markov chains. {\displaystyle \left(X_{s}:s