Editing Markov chain (section)

=== Definition ===
A Markov process is a [[stochastic process]] that satisfies the [[Markov property]] (sometimes characterized as "[[memorylessness]]"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history.<ref name=":3">{{Cite book|title=Stochastic differential equations : an introduction with applications|author=Øksendal, B. K. (Bernt Karsten) |date=2003|publisher=Springer|isbn=3540047581|edition=6th|location=Berlin|oclc=52203046}}</ref> In other words, [[conditional probability|conditional]] on the present state of the system, its future and past states are [[independence (probability theory)|independent]].

A Markov chain is a type of Markov process that has either a discrete [[state space]] or a discrete index set (often representing time), but the precise definition of a Markov chain varies.<ref name="Asmussen2003page73">{{cite book|url=https://books.google.com/books?id=BeYaTxesKy0C|title=Applied Probability and Queues|date=15 May 2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=7|author=Søren Asmussen}}</ref> For example, it is common to define a Markov chain as a Markov process in either [[continuous or discrete variable|discrete or continuous time]] with a countable state space (thus regardless of the nature of time),<ref name="Parzen1999page1882">{{cite book|url=https://books.google.com/books?id=0mB2CQAAQBAJ|title=Stochastic Processes|date=17 June 2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|page=188|author=Emanuel Parzen}}</ref><ref name="KarlinTaylor2012page292">{{cite book|url=https://books.google.com/books?id=dSDxjX9nmmMC|title=A First Course in Stochastic Processes|date=2 December 2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=29 and 30|author1=Samuel Karlin|author2=Howard E. Taylor}}</ref><ref name="Lamperti1977chap62">{{cite book|url=https://books.google.com/books?id=Pd4cvgAACAAJ|title=Stochastic processes: a survey of the mathematical theory|publisher=Springer-Verlag|year=1977|isbn=978-3-540-90275-1|pages=106–121|author=John Lamperti}}</ref><ref name="Ross1996page174and2312">{{cite book|url=https://books.google.com/books?id=ImUPAQAAMAAJ|title=Stochastic processes|publisher=Wiley|year=1996|isbn=978-0-471-12062-9|pages=174 and 231|author=Sheldon M. Ross}}</ref> but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).<ref name="Asmussen2003page73" />