Maxima and exceedances of stationary Markov chains

Recent work by Athreya and Ney and by Nummelin on the limit theory for Markov chains shows that the closed connection with regeneration theory holds also for chains on a general state space. Here this is used to study extremal behaviour of stationary (or asymptotically stationary) Markov chains. Many of the results center on the ‘clustering’ of extremes of adjacent values of the chains. In addition one criterion for convergence of extremes of general stationary sequences is derived. The results are applied to waiting times in the GI/G/1 queue and to autoregressive processes.