# Matrix of transition probabilities

The matrix $P _ {t} = \| p _ {ij} ( t) \|$ of transition probabilities in time $t$ for a homogeneous Markov chain $\xi ( t)$ with at most a countable set of states $S$:

$$p _ {ij} ( t) = {\mathsf P} \{ \xi ( t) = j \mid \xi ( 0) = i \} ,\ \ i, j \in S.$$

The matrices $\| p _ {ij} ( t) \|$ of a Markov chain with discrete time or a regular Markov chain with continuous time satisfy the following conditions for any $t > 0$ and $i, j \in S$:

$$p _ {ij} ( t) \geq 0,\ \ \sum _ {j \in S } p _ {ij} ( t) = 1,$$

i.e. they are stochastic matrices (cf. Stochastic matrix), while for irregular chains

$$p _ {ij} ( t) \geq 0,\ \ \sum _ {j \in S } p _ {ij} ( t) \leq 1,$$

such matrices are called sub-stochastic.

By virtue of the basic (Chapman–Kolmogorov) property of a homogeneous Markov chain,

$$p _ {ij} ( s+ t) = \sum _ {k \in S } p _ {ik} ( s) p _ {kj} ( t),$$

the family of matrices $\{ {P _ {t} } : {t > 0 } \}$ forms a multiplicative semi-group; if the time is discrete, this semi-group is uniquely determined by $P _ {1}$.