Namespaces
Variants
Actions

Difference between revisions of "Matrix of transition probabilities"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
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" />:
+
<!--
 +
m0628401.png
 +
$#A+1 = 13 n = 0
 +
$#C+1 = 13 : ~/encyclopedia/old_files/data/M062/M.0602840 Matrix of transition probabilities
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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" />:
+
The matrix  $  P _ {t} = \| p _ {ij} ( t) \| $
 +
of [[Transition probabilities|transition probabilities]] in time  $  t $
 +
for a homogeneous [[Markov chain|Markov chain]]  $  \xi ( t) $
 +
with at most a countable set of states  $  S $:
  
<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/m0628409.png" /></td> </tr></table>
+
$$
 +
p _ {ij} ( t)  = {\mathsf P} \{ \xi ( t) = j \mid  \xi ( 0) = i \} ,\ \
 +
i, j \in S.
 +
$$
 +
 
 +
The matrices  $  \| p _ {ij} ( t) \| $
 +
of a Markov chain with discrete time or a regular Markov chain with continuous time satisfy the following conditions for any  $  t > 0 $
 +
and  $  i, j \in S $:
 +
 
 +
$$
 +
p _ {ij} ( t)  \geq  0,\ \
 +
\sum _ {j \in S } p _ {ij} ( t)  = 1,
 +
$$
  
 
i.e. they are stochastic matrices (cf. [[Stochastic matrix|Stochastic matrix]]), while for irregular chains
 
i.e. they are stochastic matrices (cf. [[Stochastic matrix|Stochastic matrix]]), while for irregular 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/m062840/m06284010.png" /></td> </tr></table>
+
$$
 +
p _ {ij} ( t)  \geq  0,\ \
 +
\sum _ {j \in S } p _ {ij} ( t)  \leq  1,
 +
$$
  
 
such matrices are called sub-stochastic.
 
such matrices are called sub-stochastic.
Line 15: Line 41:
 
By virtue of the basic (Chapman–Kolmogorov) property of a homogeneous Markov chain,
 
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>
+
$$
 
+
p _ {ij} ( s+ t)  = \sum _ {k \in S } p _ {ik} ( s) p _ {kj} ( t),
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" />.
+
$$
 
 
  
 +
the family of matrices  $  \{ {P _ {t} } : {t > 0 } \} $
 +
forms a [[Multiplicative semi-group|multiplicative semi-group]]; if the time is discrete, this semi-group is uniquely determined by  $  P _ {1} $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.L. Chung,  "Elementary probability theory with stochastic processes" , Springer  (1974)</TD></TR></table>

Revision as of 08:00, 6 June 2020


The matrix $ P _ {t} = \| p _ {ij} ( t) \| $ of transition probabilities in time $ t $ for a homogeneous Markov chain $ \xi ( t) $ with at most a countable set of states $ S $:

$$ p _ {ij} ( t) = {\mathsf P} \{ \xi ( t) = j \mid \xi ( 0) = i \} ,\ \ i, j \in S. $$

The matrices $ \| p _ {ij} ( t) \| $ of a Markov chain with discrete time or a regular Markov chain with continuous time satisfy the following conditions for any $ t > 0 $ and $ i, j \in S $:

$$ p _ {ij} ( t) \geq 0,\ \ \sum _ {j \in S } p _ {ij} ( t) = 1, $$

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

$$ p _ {ij} ( t) \geq 0,\ \ \sum _ {j \in S } p _ {ij} ( t) \leq 1, $$

such matrices are called sub-stochastic.

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

$$ p _ {ij} ( s+ t) = \sum _ {k \in S } p _ {ik} ( s) p _ {kj} ( t), $$

the family of matrices $ \{ {P _ {t} } : {t > 0 } \} $ forms a multiplicative semi-group; if the time is discrete, this semi-group is uniquely determined by $ P _ {1} $.

Comments

References

[a1] K.L. Chung, "Elementary probability theory with stochastic processes" , Springer (1974)
How to Cite This Entry:
Matrix of transition probabilities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_of_transition_probabilities&oldid=47796
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article