The Annals of Applied Probability

Extremal Behaviour of Stationary Markov Chains with Applications

Roland Perfekt

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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.

Article information

Source
Ann. Appl. Probab., Volume 4, Number 2 (1994), 529-548.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005071

Digital Object Identifier
doi:10.1214/aoap/1177005071

Mathematical Reviews number (MathSciNet)
MR1272738

Zentralblatt MATH identifier
0806.60041

JSTOR
links.jstor.org

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60G10: Stationary processes 60G55: Point processes

Keywords
Extreme values exceedance point processes stationary Markov chains

Citation

Perfekt, Roland. Extremal Behaviour of Stationary Markov Chains with Applications. Ann. Appl. Probab. 4 (1994), no. 2, 529--548. doi:10.1214/aoap/1177005071. https://projecteuclid.org/euclid.aoap/1177005071


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