View source for Markov chain, generalized

{{MSC|60J10}}

[[Category:Markov chains]]

A sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062400/m0624001.png" /> with the properties:

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;

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" />,

<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>

<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>

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

<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>

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]].

====References====
<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>



====Comments====


====References====
<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>