## The Annals of Applied Probability

### On the distribution of tail array sums for strongly mixing stationary sequences

#### Abstract

This paper concerns the asymptotic distributions of "tail array" sums of the form $\Sigma \Psi_n (X_i - u_n)$ where ${X_i}$ is a strongly mixing stationary sequence, $\Psi_n$ are real functions which are constant for negative arguments, $\Psi_n (x) = \Psi_n (X_+)$ and ${u_n}$ are levels with $u_n \to \infty$. Compound Poisson limits for rapid convergence of $u_n \to \infty (nP{X_1 > u_n} \to \tau < \infty)$ are considered briefly and a more detailed account given for normal limits applicable to slower rates $(nP(X_1 > u_n) \to \infty)$. The work is motivated by (1) the modeling of "damage" due to very high and moderately high extremes and (2) the provision of probabilistic theory for application to problems of "tail inference" for stationary sequences.

#### Article information

Source
Ann. Appl. Probab., Volume 8, Number 3 (1998), 868-885.

Dates
First available in Project Euclid: 9 August 2002

https://projecteuclid.org/euclid.aoap/1028903454

Digital Object Identifier
doi:10.1214/aoap/1028903454

Mathematical Reviews number (MathSciNet)
MR1627799

Zentralblatt MATH identifier
0939.60007

#### Citation

Rootzén, Holger; Leadbetter, M. Ross; de Haan, Laurens. On the distribution of tail array sums for strongly mixing stationary sequences. Ann. Appl. Probab. 8 (1998), no. 3, 868--885. doi:10.1214/aoap/1028903454. https://projecteuclid.org/euclid.aoap/1028903454

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