Markov Chain

Markov Chains are described via a discrete time evolution where the transition matrix is time invariant. This is known as the Markov property or the memory-less property. This is formalised as $\{X_n{\}}_{n\in {\mathbb{Z}}^{+}}$ ,with a transition rate matrix $T$ which determines $\mathbb{P}[X_{n+1}=\alpha |X_n=\beta ]=T(\beta ,\alpha )$.