, Volume 18, Issue 3, pp 369–402 | Cite as

Statistics for tail processes of Markov chains



At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of the process at the current state and on the flow of time, either forward or backward. Estimation of the tail process provides a nonparametric approach to analyze extreme values. A duality between the distributions of the forward and backward increments provides additional information that can be exploited in the construction of more efficient estimators. The large-sample distribution of such estimators is derived via empirical process theory for cluster functionals. Their finite-sample performance is evaluated via Monte Carlo simulations involving copula-based Markov models and solutions to stochastic recurrence equations. The estimators are applied to stock price data to study the absence or presence of symmetries in the succession of large gains and losses.


Heavy-tailed Markov chains Regular variation Stationary time series Tail process Time reversibility 

AMS 2000 Subject Classifications

Primary–62G32 Secondary–62G20, 62M10 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany
  2. 2.Institut de Statistique, Biostatistique et Sciences ActuariellesUniversité Catholique de LouvainLouvain-la-NeuveBelgium

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