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 \({\mathbf \Phi}\) evolving on a space \({\mathsf X}\) and governed by an overall probability law \({\mathsf P}\) to be a time-homogeneous Markov chain there must be a set of "transition probabilities" \(\{P^n (x,A), x \in {\mathsf X}, A \subset {\mathsf X}\}\) for appropriate sets \(A\) such that for times \(n,m\) in \({\mathbb Z}_+\) (Ref. 1 Eq. 1.1)
\[{\mathsf P} (\Phi_{n+m} \in A \vert \Phi_j,j \leq m; \Phi_m =x)= P^n(x,A);\]
that is \(P^n(x,A)\) denotes the probability that a chain at x will be in the set A after n steps, or transitions. The independence of \(P^n\) on the values of \(\Phi_j,j \leq m\) is the Markov property, and the independence of \(P^n\) and m is the time-homogeneity property.
