Markov chain, class of zero states of a

2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]

A set $ K $ of states of a homogeneous 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

[C]	K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967) MR0217872 Zbl 0146.38401

Comments

Cf. also Markov chain, class of positive states of a.

References

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