View source for Matrix of transition probabilities

The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628401.png" /> of [[Transition probabilities|transition probabilities]] in time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628402.png" /> for a homogeneous [[Markov chain|Markov chain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628403.png" /> with at most a countable set of states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628404.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/m062840/m0628405.png" /></td> </tr></table>

The matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628406.png" /> of a Markov chain with discrete time or a regular Markov chain with continuous time satisfy the following conditions for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m0628408.png" />:

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i.e. they are stochastic matrices (cf. [[Stochastic matrix|Stochastic matrix]]), while for irregular chains

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such matrices are called sub-stochastic.

By virtue of the basic (Chapman–Kolmogorov) property of a homogeneous Markov chain,

<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/m062840/m06284011.png" /></td> </tr></table>

the family of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m06284012.png" /> forms a [[Multiplicative semi-group|multiplicative semi-group]]; if the time is discrete, this semi-group is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062840/m06284013.png" />.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.L. Chung,   "Elementary probability theory with stochastic processes" , Springer  (1974)</TD></TR></table>