Difference between revisions of "Markov chain, decomposable"
From Encyclopedia of Mathematics
(MSC|60J10|60J27 Category:Markov processes) |
(Tex done) |
||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | {{MSC|60J10|60J27}} | + | {{TEX|done}}{{MSC|60J10|60J27}} |
[[Category:Markov processes]] | [[Category:Markov processes]] | ||
| − | A [[ | + | A [[Markov chain]] whose [[transition probabilities]] $p_{ij}(t)$ have the following property: There are states $i,j$ such that $p_{ij}(t) = 0$ for all $t \ge 0$. Decomposability of a Markov chain is equivalent to decomposability of its matrix of transition probabilities $P = \left( {p_{ij}} \right)$ for a discrete-time Markov chain, and of its matrix of transition probability densities $Q = \left( {p'_{ij}(0)} \right)$ for a continuous-time Markov chain. 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]]). |
| Line 11: | Line 11: | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1966) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Fr}}|| 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}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|I}}|| M. Iosifescu, "Finite Markov processes and their applications", Wiley (1980) {{MR|0587116}} {{ZBL|0436.60001}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|KS}}|| J.G. Kemeny, J.L. Snell, "Finite Markov chains", v. Nostrand (1960) {{MR|1531032}} {{MR|0115196}} {{ZBL|0089.13704}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|KSK}}|| J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains", Springer (1976) {{MR|0407981}} {{ZBL|0348.60090}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Re}}|| D. Revuz, "Markov chains", North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ro}}|| V.I. Romanovsky, "Discrete Markov chains", Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0266312}} {{ZBL|0201.20002}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|S}}|| E. Seneta, "Non-negative matrices and Markov chains", Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}} | ||
| + | |} | ||
Revision as of 20:35, 29 October 2016
2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]
A Markov chain whose transition probabilities $p_{ij}(t)$ have the following property: There are states $i,j$ such that $p_{ij}(t) = 0$ for all $t \ge 0$. Decomposability of a Markov chain is equivalent to decomposability of its matrix of transition probabilities $P = \left( {p_{ij}} \right)$ for a discrete-time Markov chain, and of its matrix of transition probability densities $Q = \left( {p'_{ij}(0)} \right)$ for a continuous-time Markov chain. 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
| [F] | W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1966) |
| [Fr] | 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 |
| [I] | M. Iosifescu, "Finite Markov processes and their applications", Wiley (1980) MR0587116 Zbl 0436.60001 |
| [KS] | J.G. Kemeny, J.L. Snell, "Finite Markov chains", v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704 |
| [KSK] | J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains", Springer (1976) MR0407981 Zbl 0348.60090 |
| [Re] | D. Revuz, "Markov chains", North-Holland (1975) MR0415773 Zbl 0332.60045 |
| [Ro] | V.I. Romanovsky, "Discrete Markov chains", Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002 |
| [S] | 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