Markov chain
Properties

Two states are said to communicate with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class is closed if the probability of leaving the class is zero. A Markov chain is irreducible if there is one communicating class, the state space.

A state i has period k if k is the greatest common divisor of the number of transitions by which i can be reached, starting from i. That is:

The state is periodic if k > 1 {\displaystyle k>1}; otherwise k = 1 {\displaystyle k=1} and the state is aperiodic.

A state i is said to be transient if, starting from i, there is a non-zero probability that the chain will never return to i. It is called recurrent (or persistent) otherwise. For a recurrent state i, the mean hitting time is defined as:

State i is positive recurrent if M i {\displaystyle M_{i}} is finite and null recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties — that is, if one state has the property then all states in its communicating class have the property.

A state i is called absorbing if there are no outgoing transitions from the state.
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