Extremal Behaviour of Stationary Markov Chains with Applications

Abstract

In this paper the extremal behaviour of real-valued, stationary Markov chains is studied under fairly general assumptions. Conditions are obtained for convergence in distribution of multilevel exceedance point processes associated with suitable families of high levels. Although applicable to general stationary sequences, these conditions are tailored for Markov chains and are seen to hold for a large class of chains. The extra assumptions used are that the marginal distributions belong to the domain of attraction of some extreme value law together with rather weak conditions on the transition probabilities. Also, a complete convergence result is given. The results are applied to an AR(1) process with uniform margins and to solutions of a first order stochastic difference equation with random coefficients.