periodicity of a Markov chain
Given any state , define the set
It is not hard to see that if , then . The period of , denoted by , is defined as
A state is said to be aperiodic if . A Markov chain is called aperiodic if every state is aperiodic.
Property. If states communicate (http://planetmath.org/MarkovChainsClassStructure), then .
We will employ a common inequality involving the -step transition probabilities:
for any and non-negative integers .
Suppose first that . Since , and for some . This implies that , which forces or , and hence .
Next, assume , this means that . Since , there are such that and , and so , showing . If we pick any , we also have , or . But this means divides both and , and so divides their difference, which is . Since is arbitrarily picked, . Similarly, . Hence . ∎
|Title||periodicity of a Markov chain|
|Date of creation||2013-03-22 16:24:28|
|Last modified on||2013-03-22 16:24:28|
|Last modified by||CWoo (3771)|
|Defines||period of a state|
|Defines||aperiodic Markov chain|