Transition Probability Matrix

Definition

$P=[P_{ij}]$ A transition probability matrix $P$ is a matrix whose $(i,j)$-th element is $p_{ij}$. where each $p_{ij}$ is a Transition Probability of HMC $\{X_t{\}}_{t\ge 0}$ having countable state space

Facts

Since the number of elements of state space can be infinity, strictly speaking $P$ is not a matrix. However, several operations of matrix are satisfied for the $P$

${\sum }_{j\in E}{p}_{ij}=1$ The sum of each row of a transition matrix is one