Matrix of transition probabilities
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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
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