Matrix of transition probabilities

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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 .



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