## Bernoulli

• Bernoulli
• Volume 19, Number 4 (2013), 1419-1448.

### Clustering of Markov chain exceedances

#### Abstract

The tail chain of a Markov chain can be used to model the dependence between extreme observations. For a positive recurrent Markov chain, the tail chain aids in describing the limit of a sequence of point processes $\{N_{n},n\geq1\}$, consisting of normalized observations plotted against scaled time points. Under fairly general conditions on extremal behaviour, $\{N_{n}\}$ converges to a cluster Poisson process. Our technique decomposes the sample path of the chain into i.i.d. regenerative cycles rather than using blocking argument typically employed in the context of stationarity with mixing.

#### Article information

Source
Bernoulli, Volume 19, Number 4 (2013), 1419-1448.

Dates
First available in Project Euclid: 27 August 2013

https://projecteuclid.org/euclid.bj/1377612859

Digital Object Identifier
doi:10.3150/12-BEJSP08

Mathematical Reviews number (MathSciNet)
MR3102909

Zentralblatt MATH identifier
1284.60106

#### Citation

Resnick, Sidney I.; Zeber, David. Clustering of Markov chain exceedances. Bernoulli 19 (2013), no. 4, 1419--1448. doi:10.3150/12-BEJSP08. https://projecteuclid.org/euclid.bj/1377612859

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