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

`X`of random variables taking values in a "state space", the value of

_{1},X_{2},X_{3},....`X`being "the state of the system at time

_{n}`n`". The (discrete-time)

**Markov property**says that the conditional distribution of the "future"

`X`, depends on the past

_{1},...,X_{n}**only**through

`X`. 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

_{n}`j`"tomorrow" given that it is in state

`i`"today". The

`ij`entry in the

`k`th 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 a stochastic matrix if and only if it is the matrix of transition probabilities of some Markov chain.

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## 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. Furthemore, the concept of Markov chains has been used in bioinformatics as well. An example is the genemark algorithm for coding region/gene prediction.

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).

## See also

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