Clustering of Markov Chain Exceedances

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 ${N_n, n \geq 1}$, consisting of normalized observations plotted against scaled time points. Under fairly general conditions on extremal behavior, ${N_n}$ converges to a cluster Poisson process. Our technique decomposes the sample path of the chain into iid regenerative cycles rather than using blocking argument typically employed in the context of stationarity with mixing.