# Matrix of transition probabilities

From Encyclopedia of Mathematics

The matrix of transition probabilities in time for a homogeneous Markov chain with at most a countable set of states :

The matrices of a Markov chain with discrete time or a regular Markov chain with continuous time satisfy the following conditions for any and :

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

such matrices are called sub-stochastic.

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

the family of matrices forms a multiplicative semi-group; if the time is discrete, this semi-group is uniquely determined by .

#### Comments

#### References

[a1] | K.L. Chung, "Elementary probability theory with stochastic processes" , Springer (1974) |

**How to Cite This Entry:**

Matrix of transition probabilities.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Matrix_of_transition_probabilities&oldid=13510

This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article