Difference between revisions of "Markov chain, decomposable"
From Encyclopedia of Mathematics
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1966) {{MR|0210154}} {{ZBL|0138.10207}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Freedman, "Markov chains" , Holden-Day (1975) {{MR|0686269}} {{MR|0681291}} {{MR|0556418}} {{MR|0428472}} {{MR|0292176}} {{MR|0237001}} {{MR|0211464}} {{MR|0164375}} {{MR|0158435}} {{MR|0152015}} {{ZBL|0501.60071}} {{ZBL|0501.60069}} {{ZBL|0426.60064}} {{ZBL|0325.60059}} {{ZBL|0322.60057}} {{ZBL|0212.49801}} {{ZBL|0129.30605}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980) {{MR|0587116}} {{ZBL|0436.60001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) {{MR|1531032}} {{MR|0115196}} {{ZBL|0089.13704}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976) {{MR|0407981}} {{ZBL|0348.60090}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0266312}} {{ZBL|0201.20002}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}} </TD></TR></table> |
Revision as of 10:31, 27 March 2012
2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]
A Markov chain whose transition probabilities 
have the following property: There are states 
such that 
for all 
. Decomposability of a Markov chain is equivalent to decomposability of its matrix of transition probabilities 
for discrete-time Markov chains, and of its matrix of transition probability densities 
, 
, for continuous-time Markov chains. The state space of a decomposable Markov chain consists either of inessential states or of more than one class of communicating states (cf. Markov chain).
Comments
References
| [a1] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966) MR0210154 Zbl 0138.10207 |
| [a2] | D. Freedman, "Markov chains" , Holden-Day (1975) MR0686269 MR0681291 MR0556418 MR0428472 MR0292176 MR0237001 MR0211464 MR0164375 MR0158435 MR0152015 Zbl 0501.60071 Zbl 0501.60069 Zbl 0426.60064 Zbl 0325.60059 Zbl 0322.60057 Zbl 0212.49801 Zbl 0129.30605 |
| [a3] | M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980) MR0587116 Zbl 0436.60001 |
| [a4] | J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704 |
| [a5] | J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976) MR0407981 Zbl 0348.60090 |
| [a6] | D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045 |
| [a7] | V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002 |
| [a8] | E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) MR2209438 Zbl 0471.60001 |
How to Cite This Entry:
Markov chain, decomposable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_decomposable&oldid=21651
Markov chain, decomposable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_decomposable&oldid=21651
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article