# Rh/Fragment 031 21

## < Rh

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 Typus Verschleierung Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 31, Zeilen: 21-27
Quelle: Smith und Weissman 1996
Seite(n): 2-3, Zeilen: S.2, 29-30 - S.3, 1ff
Just as in one dimension it is the key parameter relating the extreme value properties of a stationary process to those of independent random vectors from the same d-dimensional marginal distribution. However, unlike the one dimensional case, it is not a constant for the whole process, but instead dependences on the vector τ. Some elementary properties include:

(1) $0 \le \theta(\tau) \le 1$ for all τ.

(2) For each $j= 1,\dots d,$ Xij has extremal index $\theta_j = \lim_{i_{i\ne j}\to 0^+} \theta(\tau_1,\dots,\tau_d).$

(3) $\theta(c\tau) = \theta(\tau)$ for all c > 0 (Theorem 1.1 of Nandagopalan (1994))

[Seite 2]

Just as in one dimension, it is the key parameter relating the extreme-value properties of a stationary process to those of independent random

[Seite 3]

vectors from the same D-dimensional marginal distribution. However, unlike the one-dimensional case, it is not a constant for the whole process, but instead depends on the vector τ. Some elementary properties include

(i) $0 \le \theta(\tau) \le 1$ for all τ,

(ii) if τd > 0 but τd' = 0 for all $d' \ne d,$ then $\theta(\tau) = \theta_d,$ the extremal index for the dth component process; namely $\theta(0,\dots, 0,\tau_d,0,\dots, 0) = \theta_d$

(iii) $\theta(c\tau) = \theta(\tau)$ for all c > 0 (Theorem 1.1 of Nandagopalan, 1994)

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