Let be a sequence of discrete finite random variables. The entropy rate is defined as
In general doesn't exist. Here we shall give conditions for evaluating its value. This is based on well known statistical concepts famous as stationary random variables and Markov Chain.
Definition 1.1. (Stationary). A sequence of RV's is said to be stationary if the joint probability distributions are invariant under the transition of time origin i.e.,
Definition 1.2. (Markov Chain) A sequence of random variables is said to be Markov Chain if for every N, the RV is conditionally independent of given i.e.,
Based on the above definitions the following property holds.
Property 1.62. If the sequence of random variables is stationary, we have
Moreover, the following property holds.
Property 1.63. The entropy rate of a stationary Markov sequence is given by
Note 1.5. The expression
given in the property 1.63 is famous as "Markov Entropy".