Applied Probability Trust (5 April 2016) EXTREME EVENTS OF MARKOV CHAINS

The extremal behaviour of a Markov chain is typically characterized by its tail chain. For asymptotically dependent Markov chains existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region and for asymptotically independent chains recent results fail to cover well-known asymptotically independent processes such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including an extension of the canonical Heffernan-Tawn normalization scheme, and reveal features which existing methods reduce to a degenerate form associated with non-extreme states. ∗ Postal address: University of Edinburgh, School of Mathematics, Edinburgh EH9 3FD, UK ∗ Email address: [email protected] ∗∗ Postal address: University of Mannheim, Institute of Mathematics, 68131 Mannheim, Germany ∗∗ Email address: [email protected] ∗∗∗ Postal address: Lancaster University, Department of Mathematics and Statistics, Lancaster LA1 4YF, UK ∗∗∗ Email address: [email protected] ∗∗∗∗ Postal address: Biomathematics and Statistics Scotland, Edinburgh EH9 3FD, UK ∗∗∗∗ Email address: [email protected] 1 2 Papastathopoulos, I., Strokorb, K., Tawn, A. and Butler, A.