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 evolving on a space and governed by an overall probability law to be a time-homogeneous Markov chain there must be a set of "transition probabilities" Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{P^{n}(x,A),x\in {\mathsf {X}},A\subset {\mathsf {X}}\}} for appropriate sets such that for times Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n,m} in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathbb {Z} }_{+}} (Ref. 1 Eq. 1.1)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathsf {P}}(\Phi _{n+m}\in A\vert \Phi _{j},j\leq m;\Phi _{m}=x)=P^{n}(x,A);}
that is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P^{n}} on the values of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi _{j},j\leq m} is the Markov property, and the independence of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P^{n}} and m is the time-homogeneity property.