Editing Markov chain (section)

=== Types of Markov chains ===
The system's [[state space]] and time parameter index need 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 v. continuous time:
{| class="wikitable" style="width: 60%;"
! scope="col" |
! scope="col" |Countable state space
! scope="col" |Continuous or general state space
|-
! scope="row" |Discrete-time
|(discrete-time) Markov chain on a countable or finite state space
|[[Markov chains on a measurable state space|Markov chain on a measurable state space]] (for example, [[Harris chain]])
|-
! scope="row" style="width: 10%;" |Continuous-time
|style="width: 25%;" |Continuous-time Markov process or Markov jump process
|style="width: 25%;" |Any [[continuous stochastic process]] with the Markov property (for example, the [[Wiener process]])
|}
Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a '''discrete-time Markov chain (DTMC)''',<ref name="Everitt, B.S. 2002">Everitt, B.S. (2002) ''The Cambridge Dictionary of Statistics''. CUP. {{ISBN|0-521-81099-X}}</ref> but a few authors use the term "Markov process" to refer to a '''continuous-time Markov chain (CTMC)''' without explicit mention.<ref>Parzen, E. (1962) ''Stochastic Processes'', Holden-Day. {{ISBN|0-8162-6664-6}} (Table 6.1)</ref><ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|0-19-920613-9}} (entry for "Markov chain")</ref><ref>Dodge, Y. ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|0-19-920613-9}}</ref> 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.

While the time parameter is usually discrete, the [[state space]] of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.<ref>Meyn, S. Sean P., and Richard L. Tweedie. (2009) ''Markov chains and stochastic stability''. Cambridge University Press. (Preface, p. iii)</ref> However, many applications of Markov chains employ finite or [[countable set|countably infinite]] state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see [[#Variations|Variations]]). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.