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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