View source for Markov chain, generalized

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)</TD></TR></table>



====Comments====


====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>