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In [[probability theory]] and [[statistics]], a '''Markov chain''' or '''Markoff chain''', named after the Russian mathematician [[Andrey Markov]], is a [[stochastic process]] that satisfies the [[Markov property]] (usually characterized as "[[memorylessness]]").

== Description ==

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 probability|conditional]] on the present state of the system, its future and past are [[Independence (probability theory)|independent]].

In [[Discrete time and continuous time|discrete time]], the process is known as a discrete-time Markov chain (DTMC). It undergoes transitions from one state to another on a [[state space]], with the [[probability distribution]] of the next state depending only on the current state and not on the sequence of events that preceded it.

In continuous time, the process is known as a Continuous-time Markov chain (CTMC or continuous-time Markov process). It takes values in some finite state space and the time spent in each state takes non-negative real values and has an exponential distribution. Future behavior of the model (both remaining time in current state and next state) depends only on the current state of the model and not on historical behavior.

Markov chains have many applications as [[Statistical model|statistical models]] of real-world processes.

== See also ==

* [[Brownian motion]]
* [[Dynamics of Markovian particles]]
* [[Examples of Markov chains]]
* [[Hidden Markov model]]
* [[Interacting particle system]]
* [[Markov blanket]]
* [[Markov chain approximation method]]
* [[Markov chain geostatistics]]
* [[Markov chain mixing time]]
* [[Markov chain Monte Carlo]]
* [[Markov decision process]]
* [[Markov information source]]
* [[Markov model]]
* [[Markov network]]
* [[Markov process]]
* [[Quantum Markov chain]]
* [[Random walk]]
* [[Semi-Markov process]]
* [[Stochastic cellular automaton]]
* [[Stochastic process]]
* [[Telescoping Markov chain]]
* [[Variable-order Markov model]]

== External links ==

* [https://en.wikipedia.org/wiki/Markov_chain Markov chain] @ Wikipedia.org 


[[Category:Mathematics]]