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Markov chain

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A Markov chain (named in honor of Andrei Andreevich Markov) is a stochastic process with what is called the Markov property[?], of which there is a "discrete-time" version and a "continuous-time" version. In the discrete-time case, the process consists of a sequence X1,X2,X3,.... of random variables taking values in a "state space", the value of Xn being "the state of the system at time n". The (discrete-time) Markov property says that the conditional distribution of the "future"
<math>X_{n+1}, X_{n+2}, X_{n+3},\dots</math>
given the "past", X1,...,Xn, depends on the past only through Xn. In other words, knowledge of the most recent past state of the system renders knowledge of less recent history irrelevant. Each particular Markov chain may be identified with its matrix of "transition probabilities", often called simply its transition matrix. The entries in the transition matrix are given by
<math>p_{ij}=P(X_{n+1}=j\mid X_n=i)</math>
= the probability that the system will be in state j "tomorrow" given that it is in state i "today". The ij entry in the kth power of the matrix of transition probabilities is the conditional probability that k "days" in the future the system will be in state j, given that it is in state i "today". A matrix is the matrix of transition probabilities of some Markov chain if and only if it is a stochastic matrix.

Markov chains have many scientific applications.

Markov chains are used to model various processes in queuing theory and statistics, and can also be used as a signal model in entropy coding techniques such as arithmetic coding. Markov chains also have many biological applications, particularly population processes, which are useful in modelling processes that are (at least) analogous to biological populations.

Markov processes can also be used to generate superficially "real-looking" text given a sample document: they are used in various pieces of recreational "parody generator" software (see Jeff Harrison).

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