View source for Markov chain, class of zero states of a

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{{MSC|60J10|60J27}}

[[Category:Markov processes]]

A set  $  K $
of states of a homogeneous [[Markov chain|Markov chain]]  $  \xi ( t) $
with state space  $  S $
such that

$$ 
{\mathsf P} \{ {\exists t > 0 } : {\xi ( t) = j \mid  \xi ( 0) = i } \}
 =  1
$$

for any  $  i , j \in K $,

$$ 
p _ {il} ( t)  = \ 
{\mathsf P} \{ \xi ( t) = l \mid  \xi ( 0) = i \}  =  0
$$

for any  $  i \in K $, 
$  l \in S \setminus  K $, 
$  t > 0 $, 
and

$$ \tag{* }
{\mathsf E} \tau _ {ii}  =  \infty
$$

for any  $  i \in K $, 
where  $  \tau _ {ii} $
is the return time to the state  $  i $:

$$ 
\tau _ {ii}  =  \min \ 
\{ {t > 0 } : {\xi ( t) = i \mid  \xi ( 0) = i } \}
$$

for a discrete-time Markov chain, and

$$ 
\tau _ {ii}  =  \inf \ 
\{ {t > 0 } : {\xi ( t) = i \mid  \xi ( 0) = i , \xi ( 0 + ) \neq i } \}
$$

for a continuous-time Markov chain.

As in the case of a class of positive states (in the definition of a positive class (*) is replaced by  $  {\mathsf E} \tau _ {ii} < \infty $), 
states belonging to the same zero class have a number of common properties. For example, for any states  $  i , j $
of a zero class  $  K $,

$$ 
\lim\limits _ {t \rightarrow \infty }  p _ {ij} ( t)  =  0 .
$$

An example of a Markov chain whose states form a single zero class is the symmetric random walk on the integers:

$$ 
\xi ( 0)  =  0 ,\ \ 
\xi ( t)  =  \xi ( t - 1 ) + \eta ( t) ,\ \ 
t = 1 , 2 \dots
$$

where  $  \eta ( 1) , \eta ( 2) \dots $
are independent random variables,

$$ 
{\mathsf P} \{ \eta ( i) = 1 \} \ 
=  {\mathsf P} \{ \eta ( i) = - 1 \} \ 
=  1/2 ,\  i = 1 , 2 ,\dots .
$$

====References====
{|
|valign="top"|{{Ref|C}}|| K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967) {{MR|0217872}} {{ZBL|0146.38401}}
|}

====Comments====
Cf. also [[Markov chain, class of positive states of a|Markov chain, class of positive states of a]].

====References====
{|
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''', Wiley (1966)
|-
|valign="top"|{{Ref|Fr}}|| 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|I}}|| M. Iosifescu, "Finite Markov processes and their applications", Wiley (1980) {{MR|0587116}} {{ZBL|0436.60001}}
|-
|valign="top"|{{Ref|KS}}|| J.G. Kemeny, J.L. Snell, "Finite Markov chains", v. Nostrand (1960) {{MR|1531032}} {{MR|0115196}} {{ZBL|0089.13704}}
|-
|valign="top"|{{Ref|KSK}}|| J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains", Springer (1976) {{MR|0407981}} {{ZBL|0348.60090}}
|-
|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}}
|-
|valign="top"|{{Ref|S}}|| E. Seneta, "Non-negative matrices and Markov chains", Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}}
|}