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Markov process
In probability theory and statistics, a Markov process or Markoff process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property. A Markov process can be thought of as 'memoryless': loosely speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history. i.e., conditional on the present state of the system, its future and past are independent.〔(Markov process (mathematics) ) - Britannica Online Encyclopedia〕
==Introduction==
A Markov process is a stochastic model that has the Markov property. It can be used to model a random system that changes states according to a transition rule that only depends on the current state. This article describes the Markov process in a very general sense, which is a concept that is usually specified further. Particularly, the system's state space and time parameter index needs to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time vs. continuous time.
Note that there is no definitive agreement in literature on the use of some of the terms that signify special cases of Markov processes. For example, often the term "Markov chain" is used to indicate a Markov process which has a finite or countable state-space, but Markov chains on a general state space fall under the same description. Similarly, a Markov chain would usually be defined for a discrete set of times (i.e. a discrete-time Markov chain)〔Everitt,B.S. (2002) ''The Cambridge Dictionary of Statistics''. CUP. ISBN 0-521-81099-X〕 although some authors use the same terminology where "time" can take continuous values.〔Dodge, Y. ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9〕 In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see Markov model). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term.
Markov processes arise in probability and statistics in one of two ways. A stochastic process, defined via a separate argument, may be shown mathematically to have the Markov property, and as a consequence to have the properties that can be deduced from this for all Markov processes. Alternately, in modelling a process, one may assume the process to be Markov, and take this as the basis for a construction. In modelling terms, assuming that the Markov property holds is one of a limited number of simple ways of introducing statistical dependence into a model for a stochastic process in such a way that allows the strength of dependence at different lags to decline as the lag increases.
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