  The tail chain of a Markov chain can be used to model the dependence between
extreme observations. For a positive recurrent Markov chain, the tail chain
aids in describing the limit of a sequence of point processes $\{N_n,n\geq1\}$,
consisting of normalized observations plotted against scaled time points. Under
fairly general conditions on extremal behaviour, $\{N_n\}$ converges to a
cluster Poisson process. Our technique decomposes the sample path of the chain
into i.i.d. regenerative cycles rather than using blocking argument typically
employed in the context of stationarity with mixing.
