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A sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624001.png" /> with the properties:
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1) the set of values of each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624002.png" /> is finite or countable;
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{{TEX|done}}
  
2) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624003.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624004.png" />,
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{{MSC|60J10}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624005.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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[[Category:Markov chains]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624006.png" /></td> </tr></table>
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A sequence of random variables  $  \xi _ {n} $
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with the properties:
  
A generalized Markov chain satisfying (*) is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624008.png" />-generalized. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624009.png" />, (*) is the usual [[Markov property|Markov property]]. The study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m06240010.png" />-generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m06240011.png" /> whose values are in one-to-one correspondence with the values of the vector
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1) the set of values of each  $  \xi _ {n} $
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is finite or countable;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m06240012.png" /></td> </tr></table>
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2) for any  $  n $
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and any  $  i _ {0} \dots i _ {n} $,
  
The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m06240013.png" /> forms an ordinary [[Markov chain|Markov chain]].
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$$ \tag{* }
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{\mathsf P} \{ \xi _ {n} = i _ {n} \mid
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\xi _ {0} = i _ {0} \dots \xi _ {n-} s = i _ {n-} s \dots
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\xi _ {n-} 1 = i _ {n-} 1 \} =
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$$
  
====References====
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$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Doob,  "Stochastic processes" , Wiley (1953)</TD></TR></table>
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= \
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{\mathsf P} \{ \xi _ {n} = i _ {n} \mid  \xi _ {n-} s
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= i _ {n-} s \dots \xi _ {n-} 1 = i _ {n-} 1 \} .
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$$
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A generalized Markov chain satisfying (*) is called  $  s $-
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generalized. For  $  s = 1 $,
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(*) is the usual [[Markov property|Markov property]]. The study of  $ s $-
 +
generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables  $ \eta _ {n} $
 +
whose values are in one-to-one correspondence with the values of the vector
  
 +
$$
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( \xi _ {n-} s+ 1 , \xi _ {n-} s+ 2 \dots \xi _ {n} ) .
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$$
  
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The sequence  $  \eta _ {n} $
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forms an ordinary [[Markov chain|Markov chain]].
 +
 +
====References====
 +
{|
 +
|valign="top"|{{Ref|D}}|| J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}}
 +
|}
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Freedman,   "Markov chains" , Holden-Day (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.G. Kemeny,   J.L. Snell,   "Finite Markov chains" , v. Nostrand (1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Revuz,   "Markov chains" , North-Holland (1975)</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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"E. Seneta,   "Non-negative matrices and Markov chains" , Springer (1981)</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)</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}}
 +
|-
 +
|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}}
 +
|-
 +
|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}}
 +
|}

Latest revision as of 07:59, 6 June 2020


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

A sequence of random variables $ \xi _ {n} $ with the properties:

1) the set of values of each $ \xi _ {n} $ is finite or countable;

2) for any $ n $ and any $ i _ {0} \dots i _ {n} $,

$$ \tag{* } {\mathsf P} \{ \xi _ {n} = i _ {n} \mid \xi _ {0} = i _ {0} \dots \xi _ {n-} s = i _ {n-} s \dots \xi _ {n-} 1 = i _ {n-} 1 \} = $$

$$ = \ {\mathsf P} \{ \xi _ {n} = i _ {n} \mid \xi _ {n-} s = i _ {n-} s \dots \xi _ {n-} 1 = i _ {n-} 1 \} . $$

A generalized Markov chain satisfying (*) is called $ s $- generalized. For $ s = 1 $, (*) is the usual Markov property. The study of $ s $- generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables $ \eta _ {n} $ whose values are in one-to-one correspondence with the values of the vector

$$ ( \xi _ {n-} s+ 1 , \xi _ {n-} s+ 2 \dots \xi _ {n} ) . $$

The sequence $ \eta _ {n} $ 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=13364
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article