Extremal behaviour of stationary Markov chains with applications
The paper concerns the extremal behaviour of real-valued, stationary Markov chains. The author finds conditions for convergence in distribution of the multilevel exceedance point processes for general stationary sequences. These conditions are convenient for applications to Markov chains. The results obtained extend the one-level result of \textit{H. Rootzén} [Adv. Appl. Probab. 20, No. 2, 371-390 (1988; Zbl 0654.60023)] to the multilevel case and to complete convergence. Next, the author uses the results together with ideas from a paper by \textit{R. L. Smith} [J. Appl. Probab. 29, No. 1, 37-45 (1992; Zbl 0759.60059)] to investigate extremal properties of a large class of stationary Markov chains. Finally, he presents two examples which deal with an AR(1) process with uniform margins and with solutions of a first order stochastic difference equation with random coefficients.
conditions for convergence in distribution
scientific article; zbMATH DE number 639117