# Markov chain

The concept of a Markov chain was developed by Andrey Andreyevich Markov. A Markov chain is a sequence of random variables with the property that it is forgetful of all but its immediate past. For a process ${\displaystyle {\mathbf {\Phi } }}$ evolving on a space ${\displaystyle {\mathsf {X}}}$ and governed by an overall probability law ${\displaystyle {\mathsf {P}}}$ to be a time-homogeneous Markov chain there must be a set of "transition probabilities" ${\displaystyle \{P^{n}(x,A),x\in {\mathsf {X}},A\subset {\mathsf {X}}\}}$ for appropriate sets ${\displaystyle A}$ such that for times ${\displaystyle n,m}$ in ${\displaystyle {\mathbb {Z} }_{+}}$ (Ref. 1 Eq. 1.1)
${\displaystyle {\mathsf {P}}(\Phi _{n+m}\in A\vert \Phi _{j},j\leq m;\Phi _{m}=x)=P^{n}(x,A);}$
that is ${\displaystyle P^{n}(x,A)}$ denotes the probability that a chain at x will be in the set A after n steps, or transitions. The independence of ${\displaystyle P^{n}}$ on the values of ${\displaystyle \Phi _{j},j\leq m}$ is the Markov property, and the independence of ${\displaystyle P^{n}}$ and m is the time-homogeneity property.