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Difference between revisions of "Markov chain, generalized"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}} </TD></TR></table>
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|valign="top"|{{Ref|D}}|| J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}}
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<table><TR><TD valign="top">[a1]</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">[a2]</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">[a3]</TD> <TD valign="top"> D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}} </TD></TR><TR><TD valign="top">[a4]</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">[a5]</TD> <TD valign="top"> E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , '''1–2''' , Gordon &amp; Breach (1965–1968) (Translated from French) {{MR|}} {{ZBL|0185.44502}} {{ZBL|0159.45802}} </TD></TR></table>
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|valign="top"|{{Ref|F}}|| 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}}
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|valign="top"|{{Ref|KS}}|| J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) {{MR|1531032}} {{MR|0115196}} {{ZBL|0089.13704}}
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|valign="top"|{{Ref|Re}}|| D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}}
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|valign="top"|{{Ref|Ro}}|| V.I. Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0266312}} {{ZBL|0201.20002}}
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|valign="top"|{{Ref|S}}|| E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}}
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|valign="top"|{{Ref|BF}}|| A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , '''1–2''' , Gordon &amp; Breach (1965–1968) (Translated from French) {{MR|}} {{ZBL|0185.44502}} {{ZBL|0159.45802}}
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Revision as of 06:24, 14 May 2012

2020 Mathematics Subject Classification: Primary: 60J10 [MSN][ZBL]

A sequence of random variables with the properties:

1) the set of values of each is finite or countable;

2) for any and any ,

(*)

A generalized Markov chain satisfying (*) is called -generalized. For , (*) is the usual Markov property. The study of -generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables whose values are in one-to-one correspondence with the values of the vector

The sequence forms an ordinary Markov chain.

References

[D] J.L. Doob, "Stochastic processes" , Wiley (1953) MR1570654 MR0058896 Zbl 0053.26802

Comments

References

[F] 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
[KS] J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704
[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
[BF] A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , 1–2 , Gordon & Breach (1965–1968) (Translated from French) Zbl 0185.44502 Zbl 0159.45802
How to Cite This Entry:
Markov chain, generalized. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_generalized&oldid=23624
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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