A New Approach to the Limit Theory of Recurrent Markov Chains

Let <!-- MATH $\{ {X_n};\,n \geqslant 0\}$ --> be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times <!-- MATH $\{ {N_i};\,i \geqslant 1\}$ --> such that <!-- MATH $\{ {X_{{N_i}}};{\text{ }}i \geqslant 1\}$ --> are independent and identically distributed. This idea is used to show that <!-- MATH $\{ {X_n}\}$ --> is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.